1. Quadratic Functions Sections 8.1-8.3

2. Quadratic Functions: 8.1 A quadratic function is a function that can be written in standard form: y = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0 Every quadratic function has a U-shaped graph called a parabola. If a is positive, the parabola opens up. If a is negative, the parabola opens down.

3. Quadratic Functions: 8.1 The vertex is the lowest point of a parabola that opens up or the highest point of a parabola that opens down. If the parabola opens up, the y-coordinate of the vertex is called the minimum. If the parabola opens down, the y-coordinate of the vertex is called the maximum.

4. Quadratic Functions: 8.1 The domain of a quadratic function is all real numbers. You can find the range of a quadratic function by looking at its graph. y ≥ # if the parabola opens up y ≤ # if the parabola opens down Range: y ≥ 1 Range: y ≤ 4

5. Quadratic Functions: 8.2 The solutions to a quadratic equation are called zeros Zeros are also called: Solutions, roots, x-intercepts The zeros are found by: Find the value of x when y = 0 (when you have the equation) Find the x-intercept(s) (when you have the graph) A quadratic function can have:

6. Quadratic Functions: 8.2 Axis of Symmetry: a vertical line that passes through the vertex divides the parabola into two symmetric parts.

7. Quadratic Functions: 8.2 The equation of the axis of symmetry is The vertex is a POINT (x, y). The x-coordinate of the vertex is The y-coordinate of the vertex is found by substituting the x-value into the quadratic function and ‘doing the math’.

8. Quadratic Functions: Practice 8.1-8.2 y = -2x 2 + 4x + 3 Does the parabola open up or down? What is the axis of symmetry? What is the x-coordinate of the vertex? What is the y-coordinate of the vertex? down x = 1 1 5 The vertex is the point (1,5)

9. Quadratic Functions: Practice 8.1-8.2 y = x 2 – 6x + 7 Does the parabola open up or down? What is the axis of symmetry? What is the x-coordinate of the vertex? What is the y-coordinate of the vertex? up x = 3 3 -2 The vertex is the point (3, -2)

10. Homework: 8.1 and 8.2 8.1 Page 527-528 #30-38, 45-50 8.2 Page 536 #19-32 all

11. Quadratic Functions: 8.3 Steps to graphing a quadratic function: 1.If necessary, make sure the function is in standard form. 2.Find and draw the axis of symmetry. 3.Find and plot the vertex. 4.Find and plot the y-intercept and 1 other point.

12. Quadratic Functions: 8.3 5. Reflect the two points. 6. You should SEE the parabola. Connect the points with a smooth curve. Remember the arrows at the ends.

13. Quadratic Functions: Practice 8.3 Graph each function. Label the vertex, axis of symmetry, and y-intercept. y = x 2 – 2x – 3 y = −2x 2 − 8x − 4

14. Quadratic Functions: 8.3 To solve a quadratic equation by graphing, simply GRAPH the equation and find the x-intercepts.