Quadratics

Graphing Quadratic Functions

A quadratic function is a function of the form y = ax ² + bx + c , where a≠ 0 , and a , b , and c are real numbers.

Intercepts of a Quadratic Function

The y -intercept is given by x = 0 : y = a(0²) + b(0) + c = c . Thus, the y -intercept is (0, c) .

The x -intercept is given by y = 0 : 0 = ax ² + bx + c . Thus, the x -intercept(s) can be found by factoring or by using the quadratic formula.

In addition, the discriminant gives the number of x -intercepts of a quadratic function, because it gives us the number of solutions to ax ² + bx + c = 0 . If b ² -4ac > 0 , there are 2 solutions to ax ² + bx + c = 0 and consequently 2 x -intercepts. If b ² - 4ac = 0 , there is 1 solution to ax ² + bx + c = 0 , and consequently 1 x -intercept. If b ² -4ac < 0 , there are no solutions to ax ² + bx + c = 0 , and consequently no x -intercepts. The graph of the function does not cross the x -axis; either the vertex of the parabola is above the x -axis and the parabola opens upward, or the vertex is below the x -axis and the parabola opens downward.

Completing the Square

A quadratic function in the form y = ax ² + bx + c is not always simple to graph. We do not know the vertex or the axis of symmetry simply by looking at the equation. To make the function easier to graph, we need to convert it to the form y = a(x - h)² + k . We do this by completing the square: adding and subtracting a constant to create a perfect square trinomial within our equation.

A perfect square trinomial is of the form x ² +2dx + d ² . In order to "create" a perfect square trinomial within our equation, we must find d . To find d , divide b by 2a . Then square d and multiply by a , and add and subtract ad ² to the equation (we must add and subtract in order to maintain the original equation). We now have an equation of the form y = ax ² +2adx + ad ² - ad ² + c . Factor ax ² +2adx + ad ² into a(x + d )² , and simplify - ad ² + c .

Here are the steps to completing the square, given an equation ax ² + bx + c :

1. Compute d = .
2. Add and subtract ad ² to the equation. This produces an equation of the form y = ax ² +2adx + ad ² - ad ² + c .
3. Factor ax ² +2adx + ad ² into a(x + d )² . This produces and equation of the form y = a(x + d )² - ad ² + c .
4. Simplify ad ² + c . This produces an equation of the form y = (x - h)² + k .
5. Check by plugging the point (h, k) into the original equation. It should satisfy the equation.

Example 1: Complete the square: y = x ² + 6x - 12

a = 1 , b = 6 , c = - 12

1. d = = 3
2. ad ² = 9 . y = (x ² + 6x + 9) - 9 - 12
3. y = (x + 3)² - 9 - 12
4. y = (x + 3)² - 21
5. Check: -21 = (- 3)² + 6(- 3) - 12

Example 2: Complete the square: y = 4x ² + 16x

a = 4 , b = 16 , c = 0

1. d = = 2
2. ad ² = 16 . y = (4x ² + 16x + 16) - 16
3. y = 4(x + 2)² - 16
4. y = 4(x + 2)² - 16
5. Check: -16 = 4(- 2)² + 16(- 2)

Example 3: Complete the square: y = 2x - 28x + 100

a = 2 , b = - 14 , c = 100

1. d = = - 7
2. ad ² = 98 . y = (2x - 28x + 98) - 98 + 100
3. y = 2(x - 7)² - 98 + 100
4. y = 2(x - 7)² + 2
5. Check: 2 = 2(7)² - 28(7) + 100

Example 4: Complete the square: y = - x ² + 10x - 1

a = - 1 , b = 10 , c = - 1

1. d = = - 5
2. ad ² = - 25 . y = (- x ² + 10x - 25) + 25 - 1
3. y = - (x - 5)² + 25 - 1
4. y = - (x - 5)² + 24
5. Check: 24 = - 5² + 10(5) - 1

After we complete the square, we can graph the quadratic equation using the vertex.