1. Bellwork:
Homework Check
Algebra II

2. Modeling with Quadratic Functions
2.4
Modeling with Quadratic Functions
Algebra II

3. Use the graphing calculator to make a scatter plot of the data.
Algebra II

4. Is it a quadratic or linear model? Explain.
Algebra II

5. Use the quadratic regression feature to find a quadratic model.
Algebra II

6. Graph the quadratic function on the same screen as the scatter plot to verify that it fits the data.
Algebra II

7. When does the wrench hit the ground? Explain.
Algebra II

8. What are some real-life situations that can be modeled by quadratics.
Algebra II

9. Modeling a quadratic function
When given the vertex and a point, use vertex formula:
y = a(x – h)2 + k
When given the x-intercepts (zeros), use intercept form:
y = a(x – p)(x – q)
Algebra II

10. What do you have? The vertex and a point. Use vertex form.
Example 1 1
Write a quadratic function from the graph. vertex: (2, -2) point: (4, 6) y = a(x – h)2 + k 6 = a (4 – 2)2 – 2 6 = a(2)2 – 2 6 = 4a – 2 8 = 4a a = 2 y = 2(x – 2)2 – 2
What do you have? The vertex and a point. Use vertex form.
Algebra II

11. What do you have? The vertex and a point. Use vertex form.
Example 2
Write a quadratic function from the graph. vertex: (0, 3) point: (2, 1) y = a(x – h)2 + k 1 = a(2 – 0) = a(2) = 4a - ½ = a y = -½(x)2 + 3
What do you have? The vertex and a point. Use vertex form.
Algebra II

12. Example 3 y = a(x – h)2 + k 0 = a(-3 + 1)2 – 8 0 = a(-2)2 – 8
Write a quadratic function from the given information.
Vertex is (-1, -8) and a point of (-3,0)
*vertex is h and k and point is x and y*
y = a(x – h)2 + k
0 = a(-3 + 1)2 – 8
0 = a(-2)2 – 8
0 = 4a – 8
8 = 4a
a = 2
y = 2(x + 1)2 – 8
Algebra II

13. Example 4 y = a(x – h)2 + k 7 = a(-4 + 2)2 + 5 7 = a(-2)2 + 5
Write a quadratic function from the given information.
Vertex is (-2, 5) and a point of (-4, 7)
*vertex is h and k and point is x and y*
y = a(x – h)2 + k
7 = a(-4 + 2)2 + 5
7 = a(-2)2 + 5
7 = 4a + 5
2 = 4a
a = ½
y = ½(x + 2)2 + 5
Algebra II

14. What do you have? The intercepts and a point. Use intercept form.
Example 5
Write a quadratic function from the graph. intercepts: (-4, 0), (6, 0) point: (1, 25) y = a(x – p)(x – q) 25 = a(1 + 4)(1 – 6) 25 = a(5)(-5) 25 = -25a a = -1 y = -(x + 4)(x – 6)
What do you have? The intercepts and a point. Use intercept form.
Algebra II

15. What do you have? The intercepts and a point. Use intercept form.
Example 6
Write a quadratic function from the graph. intercepts: (-3,0), (-4, 0) point: (-3½ , -¼) y = a(x – p)(x – q) - ¼= a(-3½ + 3)(-3½ + 4) -¼ = a(-½)(½) - ¼ = - ¼ a a = 1 y = 1(x + 3)(x + 4)
What do you have? The intercepts and a point. Use intercept form.
Algebra II

16. Example 7 y = a(x – p)(x – q) 84 = a(1 + 5)(1 – 8) 84 = a(6)(-7)
Write a quadratic function from the given information.
Intercepts are (-5, 0) and (8, 0) and a point of (1, 84)
*p and q are the intercepts and x and y are the point*
y = a(x – p)(x – q)
84 = a(1 + 5)(1 – 8)
84 = a(6)(-7)
84 = -42a
a = -2
y = -2(x + 5)(x – 8)
Algebra II

17. Example 8 y = a(x – p)(x – q) 27 = a(-2 – 7)(-2 – 10) 27 = a(-9)(-12)
Write a quadratic function from the given information.
Intercepts are (7, 0) and (10, 0) and a point of (-2, 27)
*p and q are the intercepts and x and y are the point*
y = a(x – p)(x – q)
27 = a(-2 – 7)(-2 – 10)
27 = a(-9)(-12)
27 = 108a
a = ¼
y = ¼(x – 7)(x – 10)
Algebra II

18. Write a linear or quadratic function.
Algebra II

19. Write a linear or quadratic function.
Algebra II

20. Write a linear or quadratic function.
Algebra II

21. Write a linear or quadratic function.
Algebra II

22. Closure: Exit Pass
Write a quadratic function in standard form with x-intercepts (4, 0) and (-2,0) that passes through point (2, -1).
Write a quadratic function in standard form whose vertex is (3, -2) and passes through (1,-7).
Algebra II