Quadratic Equations from Graphs

Write a quadratic equation given the x intercepts and one other point.

Step 1: Write the factors.

The intercepts of this function are (6, 0) and (8, 0). Write the factors.

• (x-6) and (x-8)

• (x+6) and (x+8)

• (x-6) and (x+8)

• (x+6) and (x-8)

Write a quadratic equation given the x intercepts and one other point.

Step 2: Find a.

Now that you know that the factors are (x-6) and (x-8), you must find a. Plug the coordinates of the vertex (7, 1) in for x and y and solve for a.

y = a(x-6)(x-8)

What is the value of a?

• a = 1

• a = -1

• a = 7

• a = -7

Write a quadratic equation given the x intercepts and one other point.

Step 3: Write the equation in factored form.

Now, you know that the factors are (x-6) and (x-8) and a = -1. Write the equation of the graph in factored form.

• -(x-6)(x-8)

• -(x+6)(x+8)

• (x-6)(x+8)

• (x+6)(x-8)

Now, take the factored form and write it in Standard Form (ax²+bx+c=0) by double distributing (or FOIL).

-(x-6)(x-8)=0

• x²+2x-15=0

• -x²+14x -48=0

• - x²+8x-15=0

• x² + 5x -16=0

Write a quadratic equation given the x intercepts and one other point.

Put the steps together.

1. Find the factors.
2. Solve for a by substituting in the extra point.
3. Write the equation in factored form.

The roots of the function are (-3, 0) and (5, 0).

The point (4, -3) is also on the graph.

• y=4(x+3)(x-5)

• y=3/7(x-3)(x+5)

• y=-3(x-3)(x+5)

• y=3/7(x+3)(x-5)

Write a quadratic equation in vertex form given the vertex and another point.

1. Substitute the coordinates of the vertex (1, -4) in for h and k.

Vertex form: y=a(x-h)²+k

• (x+1)²-4=y

• (x-1)²-4=y

• (x + 1)(x - 3)=y

• x2 +2x - 3=y

Convert this equation in vertex form to standard form.

(x-1)² -1

• x² -2x - 2

• x²+x-2

• x²+2

• x²-2x

Review: What are the vertex and axis of symmetry for the given graph?

• Vertex (1, -4) Axis of Symmetry: x=1

• Vertex (1, -4) Axis of Symmetry x = -4

• Vertex (-4, 1) Axis of Symmetry x=-4

• Vertex (-4, 1) Axis of Symmetry x = 1

Which parabola would open down?

• f(x)=x²

• f(x)=-x²+3x-6

• f(x) = x²-16

• f(x)=2x²-5x+7

Use any method (graph, factor, complete the square, quadratic formula) to solve the function:

f(x)=x²-7x+10

• No solutions

• x=-2 and x=-5

• x=2 and x=5

• x = 2/5