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1 Topic Drawing Graphs of Quadratic Functions Drawing Graphs of Quadratic Functions

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2 Topic Drawing Graphs of Quadratic Functions California Standards: 21.0 Students graph quadratic functions and know that their roots are the x -intercepts Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x -axis in zero, one, or two points. What it means for you: You’ll graph quadratic functions by finding their roots. Key words: quadratic parabola intercept vertex line of symmetry root

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3 Topic Drawing Graphs of Quadratic Functions In this Topic you’ll use methods for finding the intercepts and the vertex of a graph to draw graphs of quadratic functions.

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4 Topic Find the Roots of the Corresponding Equations Drawing Graphs of Quadratic Functions In general, a good way to graph the function y = ax 2 + bx + c is to find: (iii) the vertex. (ii) the y -intercept — this involves setting x = 0, (i) the x -intercepts (if there are any) — this involves solving a quadratic equation,

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5 Topic Example 1 Solution follows… Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (i) To find the x -intercepts of the graph of y = x 2 – 3 x + 2, you need to solve: x 2 – 3 x + 2 = 0 So the x -intercepts are (1, 0) and (2, 0). Using the zero property, x = 1 or x = 2. This quadratic factors to give: ( x – 1)( x – 2) = 0 Solution continues…

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6 Topic Example 1 Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (continued) (ii) To find the y -intercept, put x = 0 into y = x 2 – 3 x + 2. This gives y = 2, so the y -intercept is at (0, 2). Solution continues… (iii) The x -coordinate of the vertex is always halfway between the x -intercepts. So the x -coordinate of the vertex is given by: x = =

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7 And the y -coordinate of the vertex is: Topic Example 1 Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (continued) Solution continues… Also, the parabola’s line of symmetry passes through the vertex. So, the line of symmetry is the line x =. 3 2 So the vertex is at, – – 3 × + 2 = – y = x 2 – 3 x + 2

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8 The next function is the same as in the previous example, only multiplied by –2. Topic Example 1 Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (continued) Solution continues… y = x 2 – 3 x + 2 The coefficient of x 2 is negative this time, so the graph is concave down.

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9 Topic Example 1 Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (continued) Solution continues… y = x 2 – 3 x + 2 (i) To find the x -intercepts of the graph of y = –2 x x – 4, you need to solve: –2 x x – 4 = 0 This quadratic factors to give: –2( x – 1)( x – 2) = 0. This means the x -intercepts are at: (1, 0) and (2, 0). Using the zero property, x = 1 or x = 2.

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10 Topic Example 1 Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (continued) Solution continues… y = x 2 – 3 x + 2 (ii) Put x = 0 into y = –2 x x – 4 to find the y -intercept. The y -intercept is (0, –4).

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11 Topic Example 1 Drawing Graphs of Quadratic Functions Sketch the graphs of y = x 2 – 3 x + 2 and y = –2 x x – 4. Solution (continued) y = x 2 – 3 x + 2 (iii) The vertex is at x = –2 × + 6 × – 4 = So the y -coordinate of the vertex is at: and, the line of symmetry is the line x =. 3 2 The coordinates of the vertex are,, y = –2 x x – 4

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12 1. Find the x –intercepts (if there are any). 2. Find the y –intercepts (if there are any). 3. Find the vertex. 4. Using the vertex, x -intercepts, and y -intercepts, graph the quadratic. Topic Guided Practice Solution follows… Drawing Graphs of Quadratic Functions Exercises 1–4 are about the quadratic y = x 2 – 1. Let y = 0 and factor: 0 = ( x – 1)( x + 1) so x = 1 or x = –1. So, the x -intercepts are (1, 0) and (–1, 0). When x = 0, y = 0 – 1 = – 1. So, the y -intercept is (0, –1). x -coordinate: [1 + (–1)] ÷ 2 = 0. y -coordinate: y = 0 – 1 = –1. So, the vertex is at (0, –1). –6–4– y x –4 –6

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13 5. Find the x –intercepts (if there are any). 6. Find the y –intercepts (if there are any). 7. Find the vertex. 8. Using the vertex, x -intercepts, and y -intercepts, graph the quadratic. Topic Guided Practice Solution follows… Drawing Graphs of Quadratic Functions Exercises 5–8 are about the quadratic y = ( x – 1) 2 – 4. Rearrange to form a standard quadratic: y = x 2 – 2 x – 3 Let y = 0 and factor: 0 = ( x – 3)( x + 1) so x = 3 or x = –1. So, the x -intercepts are (3, 0) and (–1, 0). When x = 0, y = (0 – 1) 2 – 4 = 1 – 4 = –3. So, the y -intercept is (0, –3). x -coordinate: [3 + (–1)] ÷ 2 = 1. y -coordinate: y = (1 – 1) 2 – 4 = –4. So, the vertex is at (1, –4). –6–4– y x –4 –6

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14 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions For each of the quadratics in Exercises 1–2, follow these steps: i)Find the x –intercepts (if any), ii) Find the y –intercepts (if any), iii) Find the vertex, iv) Using the vertex, x -intercepts, and y -intercepts, graph the quadratic. 1. y = x 2 – 2 x 2. y = x x – 3 x -intercepts: (0, 0) and (2, 0) y -intercept: (0, 0) vertex: (1, –1) x -intercepts: (–3, 0) and (1, 0) y -intercept: (0, –3) vertex: (–1, –4) 12

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15 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions For each of the quadratics in Exercises 3–4, follow these steps: i) Find the x –intercepts (if any), ii) Find the y –intercepts (if any), iii) Find the vertex, iv) Using the vertex, x -intercepts, and y -intercepts, graph the quadratic. 3. y = –4 x 2 – 4 x y = x 2 – 4 x -intercepts: (–2, 0) and (2, 0) y -intercept: (0, –4) vertex: (0, –4) 34 x -intercepts: (–, 0) and (, 0) y -intercept: (0, 3), vertex: (–, 4)

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16 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions For each of the quadratics in Exercises 5–6, follow these steps: i) Find the x –intercepts (if any), ii) Find the y –intercepts (if any), iii) Find the vertex, iv) Using the vertex, x -intercepts, and y -intercepts, graph the quadratic. 5. y = x x y = – x x + 5 x -intercepts: (5, 0) and (–1, 0) y -intercept: (0, 5) vertex: (2, 9) 56 x -intercept: (–2, 0) y -intercept: (0, 4) vertex: (–2, 0)

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17 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions For the quadratics in Exercises 7, follow these steps: i) Find the x –intercepts (if any), ii) Find the y –intercepts (if any), iii) Find the vertex, iv) Using the vertex, x -intercepts, and y -intercepts, graph the quadratic. 7. y = –9 x 2 – 6 x x -intercepts: (–1, 0) and (, 0) y -intercept: (0, 3), vertex: (–, 4)

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18 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions Describe the characteristics of quadratic graphs of the form y = ax 2 + bx + c that have the following features, or say if they are not possible. 8. No x -intercepts 9. One x -intercept 10. Two x -intercepts 11. Three x -intercepts The graph is either concave up with the vertex above the x –axis, or concave down with the vertex below the x –axis. Not possible. The graph is either concave up with the vertex below the x –axis, or concave down with the vertex above the x –axis. The vertex is the x –intercept.

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19 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions Describe the characteristics of quadratic graphs of the form y = ax 2 + bx + c that have the following features, or say if they are not possible. 12. No y -intercepts 13. One y -intercept 14. Two y -intercepts 15. Three y -intercepts All quadratic equations of the form y = ax 2 + bx + c will have one y –intercept. Not possible.

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20 Topic Independent Practice Solution follows… Drawing Graphs of Quadratic Functions 17. Which quadratic equation has the following features? Vertex (0, 16), x -intercepts (4, 0), (–4, 0), and y -intercept (0, 16) 16. Which quadratic equation has the following features? Vertex (3, –4), x -intercepts (1, 0), (5, 0), and y -intercept (0, 5) y = ( x – 3) 2 – 4 or y = x 2 – 6 x + 5 y = – x

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21 Topic Round Up Drawing Graphs of Quadratic Functions A quadratic function has the general form y = ax 2 + bx + c (where a 0). When you draw the graph of a quadratic, the value of a determines whether the parabola is concave up (u-shaped) or concave down (n-shaped), and how steep it is. Changing the value of c moves the graph in the direction of the y -axis. Note that if a = 0, the function becomes y = bx + c, which is a linear function whose graph is a straight line.

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