4-1 Graphing Quadratic Functions

1. 4-1 Graphing Quadratic Functions Warm-up • Evaluate the expression • Find the value of x in the equation • Find the value of y in the equation • Find the value of y in the equation • Find the approximate value of y to two decimal places in the equation • –3 • x = 1 • y = 5 • y = -3/4 • y ≈ 3.47 Integrated 2 4-1 Graphing Quadratic Functions

2. 4-1 Graphing Quadratic Functions Today we will: • Understand how the coefficients of a quadratic function influence its graph • The direction it opens (up or down) • Its vertex • Its line of symmetry • Its y-intercepts Tomorrow we will: • Explore translations of parabolas. Integrated 2 4-1 Graphing Quadratic Functions

3. Parabolas Examples • The path of a jump shot as the ball travels toward the basket is a parabola. Integrated 2 4-1 Graphing Quadratic Functions

4. Key terms • Parabola – a curve that can be modeled with a quadratic function. • Quadratic function – a function that can be written in the form • Standard form of a quadratic function – the form Integrated 2 4-1 Graphing Quadratic Functions

5. Line of symmetry Line of symmetry Key terms - continued • Vertex – the point where a parabola crosses its line of symmetry. • Maximum – the vertex of a parabola that opens downward. The y-coordinate of the vertex is the maximum value of the function. • Minimum – the vertex of a parabola that opens upward. The y-coordinate of the vertex is the minimum value of the function. • y-intercept – the y-coordinate of the point where a graph crosses the y-axis. • x-intercept – the x-coordinate of the point where a graph crosses the x-axis. Integrated 2 4-1 Graphing Quadratic Functions

6. Direction and Min/Max The graph of the quadratic function , is a parabola. • If a is positive • the graph opens up • the vertex is a minimum • If a is negative • the graph opens down • the vertex is a maximum Integrated 2 4-1 Graphing Quadratic Functions

7. Line of Symmetry and Vertex • The line of symmetry is the vertical line . • The x-coordinate of the vertex is . • To find the y-coordinate of the vertex, substitute for x in the function and solve for y. • The y-intercept of the graph of a quadratic function is c. Integrated 2 4-1 Graphing Quadratic Functions

8. Example 1 Choose the function that models the parabola at the right. A. B. C. D. E. Integrated 2 4-1 Graphing Quadratic Functions

9. Example 1 Solution The graph opens down so a is negative. B & E are out. The y-intercept is –3. A is out. Find the line of symmetry. Choice C: Choice D: The line of symmetry is x = 4. C is the correct function. Integrated 2 4-1 Graphing Quadratic Functions

10. Example 2 Use the function • Tell whether the graph opens up or down. • Tell whether the vertex is a maximum or a minimum. • Find an equation for the line of symmetry. • Find the coordinates of the vertex. Integrated 2 4-1 Graphing Quadratic Functions

11. Example 2 Solution Use the function • a is positive, so the graph opens up. • The vertex is a minimum. • Equation for the line of symmetry. • Coordinates of the vertex. Integrated 2 4-1 Graphing Quadratic Functions

12. Example 3 Use the quadratic function • Without graphing, will the graph open up or down? • Is the vertex a minimum or a maximum? • What is the equation of the line of symmetry? • Find the coordinates of the vertex of the graph. • Find the y-intercept. • Graph the function. Integrated 2 4-1 Graphing Quadratic Functions

13. Example 3 Solution Use the quadratic function • The graph will open up, a is positive. • The vertex a minimum. • Equation of the line of symmetry. • Coordinates of the vertex of the graph. • The y-intercept is y = 25. • Graph the function. Integrated 2 4-1 Graphing Quadratic Functions

14. Example 3 Solution Use the quadratic function • Graph the function. y = 25 line of symmetry x = 3 Integrated 2 4-1 Graphing Quadratic Functions

15. Example 4 Use the function • Find the y-intercept of the graph. • Use a graph to estimate the x-intercepts. • Check one x-intercept by substitution. Integrated 2 4-1 Graphing Quadratic Functions

16. Example 4 Solution Use the function • Solution • The y-intercept is c or –7.75 • The x-intercepts are 2.5 and –3.1 • Check: Substitute 2.5 for x in the original equation. Integrated 2 4-1 Graphing Quadratic Functions

17. Example 5 Match each equation with its graph. 1 4 3 2 Integrated 2 4-1 Graphing Quadratic Functions

18. Website • http://www.valleyview.k12.oh.us/vvhs/dept/math/quadshelp.html Integrated 2 4-1 Graphing Quadratic Functions