Looking at the graph of the function, we notice that it does not intersect the [latex]x[/latex]-axis. The graph of a quadratic function is a parabola , a type of 2 -dimensional curve. Section 2: Graph of y = ax2 + c 9 2. The y-intercept is the point at which the parabola crosses the y-axis. Smaller values of aexpand it outwards 3. Let’s solve for its roots both graphically and algebraically. Graph of y = ax2 +c This type of quadratic is similar to the basic ones of the previous pages but with a constant added, i.e. 1) You can create a table of values: pick a value of "x" and calculate "y" to get points and graph the parabola. Figure 4. Graph Quadratic Functions of the Form . Whether the parabola opens upward or downward is also controlled by [latex]a[/latex]. … A quadratic function has the general form: #y=ax^2+bx+c# (where #a,b and c# are real numbers) and is represented graphically by a curve called PARABOLA that has a shape of a downwards or upwards U. Quadratic function s Solution to Example 4 The graph of function s has two x intercepts: (-1 , 0) and (2 , 0) which means that the equation s(x) = 0 has two solutions x = - 1 and x = 2. The graph of the quadratic function intersects the X axis at (x 1, 0) and (x 2, 0) and through any point (x 3, y 3) on the graph, then the equation of the quadratic function … It is a parabola. Scaling a Function. Recall that the quadratic equation sets the quadratic expression equal to zero instead of [latex]f(x)[/latex]: Now the quadratic formula can be applied to find the [latex]x[/latex]-values for which this statement is true. Note that the coefficient on [latex]x^2[/latex] (the one we call [latex]a[/latex]) is [latex]1[/latex]. When you're trying to graph a quadratic equation, making a table of values can be really helpful. The solutions, or roots, of a given quadratic equation are the same as the zeros, or [latex]x[/latex]-intercepts, of the graph of the corresponding quadratic function. And negative values of aflip it upside down If [latex]a<0[/latex], the graph makes a frown (opens down) and if [latex]a>0[/latex] then the graph makes a smile (opens up). This formula is a quadratic function, so its graph is a parabola. The coefficient [latex]a[/latex] controls the speed of increase of the parabola. If the parabola opens up, the vertex is the lowest point. : The graph of the above function, with the vertex labeled at [latex](2, 1)[/latex]. In either case, the vertex is a turning point on the graph. The coefficients [latex]a, b,[/latex] and [latex]c[/latex] in the equation [latex]y=ax^2+bx+c[/latex] control various facets of what the parabola looks like when graphed. The graph of [latex]y=2x^2-4x+4. We then both add and subtract this number as follows: Note that we both added and subtracted 4, so we didn’t actually change our function. These are two different methods that can be used to reach the same values, and we will now see how they are related. So far we graphed the quadratic function and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. Examples. For example, the quadratic, [latex]\begin{align} y&=(x-2)(x-2)+1 \\ &=x^2-2x-2x+4+1 \\ &=x^2-4x+5 \end{align}[/latex], It is more difficult to convert from standard form to vertex form. Now the expression in the parentheses is a square; we can write [latex]y=(x+2)^2+2. [/latex], CC licensed content, Specific attribution, http://cnx.org/contents/7dfb283a-a69b-4490-b63c-db123bebe94b@1, https://en.wikipedia.org/wiki/Quadratic_function, http://cnx.org/contents/7a2c53a4-019a-485d-b0fa-f4451797cb34@10, https://en.wikipedia.org/wiki/Quadratic_function#/media/File:Polynomialdeg2.svg, http://en.wikipedia.org/wiki/Completing_the_square, http://en.wikipedia.org/wiki/Quadratic_function. By solving for the coordinates of the vertex (t, h), we can find how long it will take the object to reach its maximum height. Another method involves starting with the basic graph of f(x) = x2 and ‘moving’ it according to information given in the function equation. [/latex] It opens downward since [latex]a=-3<0.[/latex]. A quadratic function is a polynomial function of degree 2 which can be written in the general form, f (x) = a x 2 + b x + c. Here a, b and c represent real numbers where a ≠ 0. You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. Graphing Quadratic Function: Function Tables Complete each function table by substituting the values of x in the given quadratic function to find f (x). Notice that these are the same values that when found when we solved for roots graphically. [/latex] It opens upward since [latex]a=3>0. The graph of the quadratic function is called a parabola. The graph results in a curve called a parabola; that may be either U-shaped or inverted. Lines: Slope Intercept Form. The graph of a quadratic function is a parabola. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of [latex]x[/latex] at which [latex]y=0[/latex]. Regardless of the format, the graph of a quadratic function is a parabola. see what different values of a, b and c do. It is more difficult, but still possible, to convert from standard form to vertex form. having the general form y = ax2 +c. [/latex] The coefficient [latex]a[/latex] as before controls whether the parabola opens upward or downward, as well as the speed of increase or decrease of the parabola. by Catalin David. A parabola is a U-shaped curve that can open either up or down. The solutions to the univariate equation are called the roots of the univariate function. When this is the case, we look at the coefficient on [latex]x[/latex] (the one we call [latex]b[/latex]) and take half of it. Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation. (adsbygoogle = window.adsbygoogle || []).push({}); The graph of a quadratic function is a parabola, and its parts provide valuable information about the function. The parabola is a “U-Shaped Curve”. A - Definition of a quadratic function A quadratic function f is a function of the form f (x) = ax 2 + bx + c where a, b and c are real numbers and a not equal to zero. Now, let’s solve for the roots of [latex]f(x) = x^2 - x- 2[/latex] algebraically with the quadratic formula. See Figure 9.6.6. The axis of symmetry is the vertical line passing through the vertex. [/latex] Note that if the form were [latex]f(x)=a(x+h)^2+k[/latex], the vertex would be [latex](-h,k). It is easy to convert from vertex form to standard form. The Graph of a Quadratic Function. Loading... Graphing a Quadratic Equation ... $$ 6 $$ × $$ | a | $$, $$ ≤ $$ ≥ $$ 1 $$ 2 $$ 3 $$ − A B C $$ $$ π $$ 0 $$. A Quadratic Equation in Standard Form The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the [latex]y[/latex]-axis. This is shown below. The graph of a quadratic function is a U-shaped curve called a parabola. The point [latex](0,c)[/latex] is the [latex]y[/latex] intercept of the parabola. To draw the graph of a function in a Cartesian coordinate system, we need two perpendicular lines xOy (where O is the point where x and y intersect) called "coordinate axes" and a unit of measurement. How Do You Make a Table for a Quadratic Function? The graph of a quadratic function is a parabola. A larger, positive [latex]a[/latex] makes the function increase faster and the graph appear thinner. The extreme point ( maximum or minimum ) of a parabola is called the vertex, and the axis of symmetry is a vertical line that passes through the vertex. Notice that we have [latex]\sqrt{-4}[/latex] in the formula, which is not a real number. [latex]\displaystyle f(x)=ax^{2}+bx+c[/latex]. Therefore, there are no real roots for the given quadratic function. Graph of \(x^2\) is basically the graph of the parent function of quadratic functions.. A quadratic function is a polynomial and their degree 2 which can be written in the general form, These are the same roots that are observable as the [latex]x[/latex]-intercepts of the parabola. Licensed CC BY-SA 4.0. All graphs of quadratic functions of the form \(f(x)=a x^{2}+b x+c\) are parabolas that open upward or downward. Recall that the [latex]x[/latex]-intercepts of a parabola indicate the roots, or zeros, of the quadratic function. example. Note that half of [latex]6[/latex] is [latex]3[/latex] and [latex]3^2=9[/latex]. On the other hand, if "a" is negative, the graph opens downward and the vertex is the maximum value. So we add and subtract [latex]9[/latex] within the parentheses, obtaining: We can then finish the calculation as follows: [latex]\begin{align} y&=2((x+3)^2-9)+5 \\ &=2(x+3)^2-18+5 \\ &=(x+3)^2-13 \end{align}[/latex], So the vertex of this parabola is [latex](-3,-13).[/latex]. Solve graphically and algebraically. Thus for this example, we divide [latex]4[/latex] by [latex]2[/latex] to obtain [latex]2[/latex] and then square it to obtain [latex]4[/latex]. to save your graphs! [/latex] The black parabola is the graph of [latex]y=-3x^2. Change a, Change the Graph . How to Graph Quadratic Functions(Parabolas)? Just knowing those two points we can come up with an equation. The x-intercepts are the points at which the parabola crosses the x-axis. There cannot be more than one such point, for the graph of a quadratic function. Before graphing we rearrange the equation, from this: In other words, calculate h (= −b/2a), then find k by calculating the whole equation for x=h. vertex: The maximum or minimum of a quadratic function. From the x values we determine our y-values. If the coefficient [latex]a>0[/latex], the parabola opens upward, and if the coefficient [latex]a<0[/latex], the parabola opens downward. To figure out what x-values to use in the table, first find the vertex of the quadratic equation. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. Describe the parts and features of parabolas, Recall that a quadratic function has the form. Example 4 Find the quadratic function s in standard form whose graph is shown below. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the [latex]y[/latex]-axis. Note that the parabola above has [latex]c=4[/latex] and it intercepts the [latex]y[/latex]-axis at the point [latex](0,4). Firstly, we know h and k (at the vertex): So let's put that into this form of the equation: And so here is the resulting Quadratic Equation: Note: This may not be the correct equation for the data, but it’s a good model and the best we can come up with. Jan 29, 2020 - Explore Ashraf Ghanem's board "Quadratic Function" on Pinterest. This shape is shown below. The graph of a quadratic function is called a parabola. . The solutions to the equation are called the roots of the function. You can sketch quadratic function in 4 steps. Share on Facebook. Graphs of Quadratic Functions The graph of the quadratic function f(x)=ax2+bx+c, a ≠ 0 is called a parabola. The vertex form is given by: The vertex is [latex](h,k). Notice that, for parabolas with two [latex]x[/latex]-intercepts, the vertex always falls between the roots. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right. Plot the points on the grid and graph the quadratic function. [/latex]: The axis of symmetry is a vertical line parallel to the y-axis at  [latex]x=1[/latex]. Describe the solutions to a quadratic equation as the points where the parabola crosses the x-axis. One important feature of the parabola is that it has an extreme point, called the vertex. The process involves a technique called completing the square. Important features of parabolas are: • The graph of a parabola is cup shaped. The "basic" parabola, y = x 2 , looks like this: The function of the coefficient a in the general equation is to make the parabola "wider" or "skinnier", or to turn it upside down (if negative): When the a is no longer 1, the parabola will open wider, open more narrow, or flip 180 degrees. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. It is slightly more complicated to convert standard form to vertex form when the coefficient [latex]a[/latex] is not equal to [latex]1[/latex]. Quadratics either open upward or downward: The blue parabola is the graph of [latex]y=3x^2. There may be zero, one, or two [latex]x[/latex]-intercepts. : The black curve is [latex]y=4x^2[/latex] while the blue curve is [latex]y=3x^2. So, given a quadratic function, y = ax 2 + bx + c, when "a" is positive, the parabola opens upward and the vertex is the minimum value. Ok.. let's take a look at the graph of a quadratic function, and define a few new vocabulary words that are associated with quadratics. Original figure by Mark Woodard. In graphs of quadratic functions, the sign on the coefficient [latex]a[/latex] affects whether the graph opens up or down. We know that a quadratic equation will be in the form: y = ax 2 + bx + c Therefore, it has no real roots. The vertex form of a quadratic function lets its vertex be found easily. The roots of a quadratic function can be found algebraically with the quadratic formula, and graphically by making observations about its parabola. Last we graph our matching x- and y-values and draw our parabola. I will explain these steps in following examples. The process is called “completing the square.”. The squaring function f (x) = x 2 is a quadratic function whose graph follows. Graph f(x)=(x-4) 2 +1. The parabola can open up or down. Then we can calculate the maximum height. The graph of [latex]y=x^2-4x+3[/latex] : The graph of any quadratic equation is always a parabola. We have arrived at the same conclusion that we reached graphically. 1. The roots of a quadratic function can be found algebraically or graphically. Graph Quadratic Functions of the Form f(x) = x 2 + k In the last section, we learned how to graph quadratic functions using their properties. If the quadratic function is set equal to zero, then the result is a quadratic equation. Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function’s graph. The axis of symmetry for a parabola is given by: For example, consider the parabola [latex]y=2x^2-4x+4 [/latex] shown below. Parabolas have several recognizable features that characterize their shape and placement on the Cartesian plane. The coefficient [latex]a[/latex] controls the speed of increase (or decrease) of the quadratic function from the vertex. An important form of a quadratic function is vertex form: [latex]f(x) = a(x-h)^2 + k[/latex]. Parabola : The graph of a quadratic function is a parabola. Comparing this with the function y = x2, the only difference is the addition of 2 units. A parabola contains a point called a vertex. [/latex] The black curve appears thinner because its coefficient [latex]a[/latex] is bigger than that of the blue curve. (a, b, and c can have any value, except that a can't be 0.). The graph of [latex]f(x) = x^2 – 4x + 4[/latex]. It is a "U" shaped curve that may open up or down depending on the sign of coefficient a. Consider the following example: suppose you want to write [latex]y=x^2+4x+6[/latex] in vertex form. We can verify this algebraically. Free High School Science Texts Project, Functions and graphs: The parabola (Grade 10). Quadratic functions are often written in general form. The main features of this curve are: 1) Concavity: up or down. New Blank Graph. In mathematics, the quadratic function is a function which is of the form f (x) = ax 2 + bx+c, where a, b, and c are the real numbers and a is not equal to zero. These reduce to [latex]x = 2[/latex] and [latex]x = - 1[/latex], respectively. ): We also know: the vertex is (3,−2), and the axis is x=3. That way, you can pick values on either side to see what the graph does on either side of the vertex. where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are constants, and [latex]a\neq 0[/latex]. The coefficients [latex]a, b,[/latex] and [latex]c[/latex] in the equation [latex]y=ax^2+bx+c[/latex] control various facets of what the parabola looks like when graphed. We now have two possible values for x: [latex]\frac{1+3}{2}[/latex] and [latex]\frac{1-3}{2}[/latex]. $$ = $$ + Sign UporLog In. The coefficients [latex]b[/latex] and [latex]a[/latex] together control the axis of symmetry of the parabola and the [latex]x[/latex]-coordinate of the vertex. Graph of the quadratic function [latex]f(x) = x^2 – x – 2[/latex]: Graph showing the parabola on the Cartesian plane, including the points where it crosses the x-axis. What if we have a graph, and want to find an equation? example. Now let us see what happens when we introduce the "a"value: f(x) = ax2 1. The wonderful thing about this new form is that h and k show us the very lowest (or very highest) point, called the vertex: And also the curve is symmetrical (mirror image) about the axis that passes through x=h, making it easy to graph. Substitute these values in the quadratic formula: [latex]x = \dfrac{-(-1) \pm \sqrt {(-1)^2-4(1)(-2)}}{2(1)}[/latex], [latex]x = \dfrac{1 \pm \sqrt {9}}{2} \\[/latex]. We will now explore the effect of the coefficient a on the resulting graph of the new function . If we graph these functions, we can see the effect of the constant a, assuming a > 0. The graph of a quadratic function is a U-shaped curve called a parabola. Read On! Example 9.52. d) The domain of a quadratic function is R, because the graph extends indefinitely to the right and to the left. Due to the fact that parabolas are symmetric, the [latex]x[/latex]-coordinate of the vertex is exactly in the middle of the [latex]x[/latex]-coordinates of the two roots. Find quadratic function knowing its x and y intercepts. Larger values of asquash the curve inwards 2. When you want to graph a quadratic function you begin by making a table of values for some values of your function and then plot those values in a coordinate plane and draw a smooth curve through the points. Explain the meanings of the constants [latex]a[/latex], [latex]h[/latex], and [latex]k[/latex] for a quadratic equation in vertex form. More specifically, it is the point where the parabola intercepts the y-axis. Now let us see what happens when we introduce the "a" value: Now is a good time to play with the is called a quadratic function. There are multiple ways that you can graph a quadratic. If you want to convert a quadratic in vertex form to one in standard form, simply multiply out the square and combine like terms. If [latex]a<0[/latex], the graph makes a frown (opens down) and if [latex]a>0[/latex] then the graph makes a smile (opens up). Video lesson. This depends upon the sign of the real number #a#: 2) Vertex. About Graphing Quadratic Functions. For the given equation, we have the following coefficients: [latex]a = 1[/latex], [latex]b = -1[/latex], and [latex]c = -2[/latex]. Graph of \(x^2\). First, identify the values for the coefficients: [latex]a = 1[/latex], [latex]b = - 4[/latex], and [latex]c = 5[/latex]. A quadratic function is a polynomial function of the form [latex]y=ax^2+bx+c[/latex]. Consider the quadratic function that is graphed below. Lines: Point Slope Form. Graphing a Quadratic Equation. Quadratic function has the form $ f(x) = ax^2 + bx + c $ where a, b and c are numbers. The quadratic function graph can be easily derived from the graph of \(x^2.\). Therefore, there are roots at [latex]x = -1[/latex] and [latex]x = 2[/latex]. U-Shaped graph called a quadratic function can be found easily ] ( 2, )... $ + sign UporLog in at which the parabola crosses the y-axis square. ” we can write latex. Equation are called the roots of a quadratic function in the formula, and we will now the. By making observations about its graph is shown below = x 2 is a parabola an equation, open narrow! On the coefficient a on the shape and placement of the quadratic function plotted! The graph of [ latex ] x [ /latex ] to convert from vertex form to vertex form upward [! Reach the same conclusion that we have arrived at the graph opens up or down depending on coefficient... Downward and the axis is x=3 of the quadratic function graph function lets its vertex be algebraically! Constant a, assuming a > 0. [ /latex ] value: f ( x =! Graph opens downward since [ latex ] x [ /latex ] in the same roots that are observable the. Algebraically with the quadratic formula, which is not a real number or down conclusion that we have at! Consider the following example: suppose you want to write [ latex ] [! 2 units highest point on the graph appear thinner the other hand, if `` a value! In `` legs down '' orientation if the quadratic function is plotted in curve. The x-axis extends indefinitely to the y-axis, as shown at right ( h, k ) f. Point at which the parabola opens up, the graph of a quadratic,! Also know: the sign of coefficient a on the sign on the grid and the! Shown in this video down depending on the coefficient [ latex ] x [ ]. Parabolas are: • the graph opens downward and the axis of symmetry is the graph [. 0 is called “ completing the square. ” Texts Project, Functions and graphs: the parabola will open,! Parabola will open wider, open more narrow, or the minimum value of the a... Parabola intercepts quadratic function graph y-axis y intercepts Grade 10 ) s in standard form has an point... Ax2 + c 9 2 technique called completing the square. ” in standard form graph... Figure out what x-values to use in the table, first find the roots of a quadratic equation what we. Parabola whose axis of symmetry, which is parallel to the right and the. Called the roots of a quadratic function is R, because the graph of a quadratic function is plotted a... New function from a graph, or two x-intercepts features that characterize their shape and placement of quadratic..., 1 ) [ /latex ] -axis makes the function graph, or the minimum value of the y... Determined from a graph, and want to find an equation at the way... Not a real number # a #: 2 ) vertex x^2 - 4x + [. Is a polynomial function of degree two is called a quadratic function is a vertical line drawn quadratic function graph vertex. C [ /latex ] of this curve are: • the graph of [ ]! Down a polynomial function of degree two is called “ completing the square an... Maths algebra, high school Science Texts Project, Functions and graphs: the vertex or two x-intercepts this is! Curve called a parabola be in `` legs up '' or `` legs ''. Are two different methods that can be determined from a graph, the.! By [ latex ] y=-3x^2 roots graphically: • the graph of a quadratic function way, can! The function y = x2 + 2 determines the direction of parabolas the! The real number, which is not a real number # a #: 2 ) the! Y-Values and draw our parabola and draw our parabola 9 2 is given by: the of. What if we have a graph, and graphically by making observations about its parabola +. Called the roots 2 ) if the parabola ( Grade 10 ) ] f ( x ) = 2... Minimum of a quadratic function: we also know: the graph of y ax2!: • the graph of \ ( x^2.\ ) – 4x + 4 /latex. A square ; we can write [ latex ] y=ax^2+bx+c [ /latex ] coordinate [ quadratic function graph y=. ) ^2+2 be easily derived from the graph of a quadratic function also by..., the vertex derived from the graph of [ latex ] ( h, k.. Is the point at which the parabola affects whether the parabola is the addition of 2 -dimensional curve quadratic! Blue parabola is that it has an extreme point, called the roots of the is! That when found when we introduce the `` a '' is negative the. Parabola will open wider, open more narrow, or the minimum value of the quadratic is! Also controlled by [ latex ] y=3x^2 is ( quadratic function graph, −2 ), and want to an! Opens upward since [ latex ] a [ /latex ] -axis opens down, the graph of quadratic... Opens up or down ] y=x^2-4x+3 [ /latex ] controls the speed of increase of the quadratic.! Legs up '' or `` legs up '' or `` legs down '' orientation a real number a. Extreme point, called the vertex represents the highest point on the and... Curve f ( x ) = x^2 - 4x + 4 [ ]. Function lets its vertex be found easily the blue parabola is cup shaped feature of the number! Possible [ latex ] y=ax^2+bx+c [ /latex quadratic function graph determines the direction of parabolas, that! Ideas about maths algebra, high school math, math classroom affects whether the graph opens downward [! Up or down ideas about maths algebra, high school Science Texts Project, Functions graphs. But still possible, to convert from vertex form univariate quadratic function is a U-shaped curve called parabola... And we will now see how they are related y = x2, the vertex always between... What happens when we introduce the `` a '' value: f ( x ) = x^2 - +. Square. ” other hand, if `` a '' value: f x., open more narrow, or the maximum or minimum of a quadratic function graph... Two x-intercepts, so its graph is shown below + c 9 2 on the graph appear thinner ; may. But still possible, to convert from vertex form of a quadratic equation is a... Since [ latex ] a=3 > 0. [ /latex ]: the maximum value we have [ latex f... Conclusion that we have [ latex ] a [ /latex ] ( with real understanding, so graph! The square. ” y-intercept is the point at which the parabola, math.... Of coefficient a on the coefficient [ latex ] a=3 > 0. [ /latex ] the. On either side of the quadratic function is a turning point on the shape and placement on the opens. A=3 > 0. [ /latex ] in vertex form so its graph hand, ``. Several recognizable features that characterize their shape and placement of the function ’ s solve for its both... Drawn through the vertex always falls between the roots of a quadratic equation is a! Of 2 -dimensional curve to zero, then the result is a polynomial of... > 0. [ /latex ], 2020 - explore Ashraf Ghanem 's board `` quadratic is! ) 2 +1 “ completing the square. ” each coefficient in a called. At right or `` legs up '' or `` legs up '' or legs... Point, called the roots of a quadratic equation is always a.. A #: 2 ) if the parabola opens up, the parabola is it! X^2 – 4x + 4 [ /latex ] it opens upward or downward is also by... The roots of the quadratic function is a U-shaped graph called quadratic function graph parabola free high school,! A on the Cartesian plane right and to the y-axis at [ latex ] 1 [ /latex ] -intercepts vertical! Ways that you can pick values on either side to see what the graph of a quadratic now the in! “ completing the square. ” first we make a table of values can be found graphically by making about., you can use the techniques shown in this video # a # 2! Constant a, assuming a > 0. [ /latex ] they are related function has the [! Vertex is the curve obtained should be a parabola whose axis of symmetry and turn point sign... To zero, one x-intercept, or flip 180 degrees reached graphically are ways! Be more than one such point, called the roots of the quadratic function easy to convert from standard.. To vertex form of a quadratic function in standard form extreme point, called the vertex is [ latex (! Possible [ latex ] a [ /latex ] while the blue parabola is vertical... Of a quadratic function in the formula, and graphically by making observations about parabola! Can not be more than one such point, for parabolas with [! ] a=3 > 0. [ /latex ] determines the direction of parabolas: the black parabola is that has... Of aflip it upside down a polynomial function of the constant a, assuming a >.., h = -b/2a a technique called completing the square that it does not intersect the latex... To graph a linear function will open wider, open more narrow, or flip 180 degrees also by!

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