Five-Minute Check (over Lesson 9–7) Mathematical Practices Then/Now New Vocabulary Example 1: Find the Equation of a Parabola Example 2: Real-World Example: Write an Equation of a Parabola Key Concept: Standard Form of Equations of Parabolas Example 3: Analyze and Graph a Parabola Lesson Menu

Name a radius. A. B. C. D. 5-Minute Check 1

Name a chord. A. B. C. D. 5-Minute Check 2

Name a diameter. A. B. C. D. 5-Minute Check 3

A. 90 B. 120 C. 160 D. 170 5-Minute Check 4

Write an equation of the circle with center at (–3, 2) and a diameter of 6. A. (x + 3) + (y – 2) = 9 B. (x – 3) + (y + 2) = 6 C. (x + 3)2 + (y – 2)2 = 9 D. (x – 3)2 + (y + 2)2 = 6 5-Minute Check 5

Which of the following figures is always perpendicular to a radius of a circle at their intersection on the circle? A. chord B. diameter C. secant D. tangent 5-Minute Check 6

Mathematical Practices 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. 4 Model with mathematics. 7 Look for and make use of structure. Content Standards G.GPE.2 Derive the equation of a parabola given a focus and directrix. MP

You found the equation of a circle by using the Distance Formula. Derive the equation of a parabola given a focus and directrix. Then/Now

parabola focus directrix axis of symmetry New Vocabulary

Write an equation of each parabola. Write the Equation of a Parabola Write an equation of each parabola. A. focus (0, −4) and directrix y = 4 The vertex is the midpoint between the focus and the directrix. Use the vertex to find the values of h and k. (0, 0); h = 0, k = 0 p = 4 The distance between the vertex and the focus is p. The graph points down, so use –p for the equation. Example 1

Write the Equation of a Parabola Substitute the values into the equation (x – h)2 = 4p(y – k) and simplify. (x – 0)2 = 4(–4)(y – 0) x2 = –16y Answer: Example 1

Write an equation of each parabola. Write the Equation of a Parabola Write an equation of each parabola. B. vertex (–2, 6), focus (–2, –2) Use the vertex to find the values of h and k. (–2, 6); h = –2, k = 6 The distance between the vertex and the focus is p. The graph points down, so use –p for the equation. p = 8 Example 1

Write the Equation of a Parabola Substitute the values into the equation (x – h)2 = 4p(y – k) and simplify. (x + 2)2 = 4(–8)(y – 6) (x + 2)2 = –32(y – 6) Answer: (x + 2)2 = –32(y – 6) Example 1

A parabolic telescope mirror opening upward has Write an Equation of a Parabola A parabolic telescope mirror opening upward has a diameter of 1.5 meters. The distance between the vertex and the focus is 0.8 meter. Write the equation of a cross section of the mirror. Use the vertex to find the values of h and k. (0, 0); h = 0, k = 0 The distance between the vertex and the focus is p. p = 0.8 Example 2

Write an Equation of a Parabola Substitute the values into the equation (x – h)2 = 4p(y – k) and simplify. (x – 0)2 = 4(0.8)(y – 0) x2 = 3.2y Answer: x2 = 3.2y Example 2

Key Concept

Use the standard form of a parabola to find the vertex (h, k). Analyze and Graph a Parabola Graph each equation. A. y = 2(x – 1)2 – 5 y = 2(x – 1)2 – 5 Use the standard form of a parabola to find the vertex (h, k). (1, –5) Plot the vertex. The graph points up and is narrow. a = 2 Example 3

Analyze and Graph a Parabola Sketch the graph. Answer: Example 3

Use the standard form of a parabola to find the vertex (h, k). Analyze and Graph a Parabola Graph each equation. B. –(x – 3) = (y – 2)2 –(x – 3) = (y – 2)2 Use the standard form of a parabola to find the vertex (h, k). (3, 2) Plot the vertex. Example 3

a = –1 The graph points left. Sketch the graph. Answer: Analyze and Graph a Parabola a = –1 The graph points left. Sketch the graph. Answer: Example 3