Graphing quadratic equations

1. Solving Quadratic Equations by Graphing

2. Quadratic Equation y = ax 2 + bx + c ax 2 is the quadratic term. bx is the linear term. c is the constant term. The highest exponent is two; therefore, the degree is two.

3. Example f(x)=5x 2 -7x+1 Quadratic term 5x 2 Linear term -7x Constant term 1 Identifying Terms

4. Example f(x) = 4x 2 - 3 Quadratic term 4x 2 Linear term 0 Constant term -3 Identifying Terms

5. Now you try this problem. f(x) = 5x 2 - 2x + 3 quadratic term linear term constant term Identifying Terms 5x 2 -2x 3

6. The number of real solutions is at most two. Quadratic Solutions No solutions One solution Two solutions

7. Quadratic Function y = ax 2 + bx + c Quadratic Term Linear Term Constant Term What is the linear term of y = 4x 2 – 3? 0x What is the linear term of y = x 2 - 5x ? -5x What is the constant term of y = x 2 – 5x? Can the quadratic term be zero? No!

8. Solving Equations When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts. These values are also referred to as solutions, zeros, or roots.

9. Example f(x) = x 2 - 4 Identifying Solutions Solutions are -2 and 2.

10. Now you try this problem. f(x) = 2x - x 2 Solutions are 0 and 2. Identifying Solutions

11. Quadratic Functions The graph of a quadratic function is a: A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex ( minimum ). If the parabola opens down, the vertex is the highest point ( maximum ). NOTE: if the parabola opens left or right it is not a function! y x Vertex Vertex parabola

12. y = ax 2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Standard Form y x The standard form of a quadratic function is: a > 0 a < 0

13. y x Axis of Symmetry Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the Axis of symmetry . The Axis of symmetry ALWAYS passes through the vertex. If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side.

14. Find the Axis of symmetry for y = 3 x 2 – 18 x + 7 Finding the Axis of Symmetry When a quadratic function is in standard form the equation of the Axis of symmetry is y = ax 2 + bx + c , This is best read as … ‘the opposite of b divided by the quantity of 2 times a .’ The Axis of symmetry is x = 3. a = 3 b = -18

15. Finding the Vertex The Axis of symmetry always goes through the _______ . Thus, the Axis of symmetry gives us the ____________ of the vertex. STEP 1: Find the Axis of symmetry Vertex Find the vertex of y = -2x 2 + 8x - 3 a = -2 b = 8 X-coordinate The x-coordinate of the vertex is 2

16. Finding the Vertex STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex. The vertex is (2 , 5) Find the vertex of y = -2x 2 + 8x - 3

17. Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form . STEP 1 : Find the Axis of symmetry using: STEP 2 : Find the vertex STEP 3 : Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry.

18. STEP 1 : Find the Axis of symmetry y x Graphing a Quadratic Function STEP 2 : Find the vertex Substitute in x = 1 to find the y – value of the vertex.

19. The graph of a quadratic equation is a parabola. The roots or zeros are the x-intercepts. The vertex is the maximum or minimum point. All parabolas have an axis of symmetry. Graphing Quadratic Equations

20. One method of graphing uses a table with arbitrary x-values. Graph y = x 2 - 4x Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2 Graphing Quadratic Equations x y 1 -3 2 -4 3 -3 4

21. Try this problem y = x 2 - 2x - 8. Roots Vertex Axis of Symmetry Graphing Quadratic Equations x y -2 -1 1 3 4