Graphing Quadratic Equations: example 10.53

Find the minimum value of the quadratic equation $y = x^{2} + 2 x - 8$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the maximum or minimum value of the quadratic equation $y = x^{2} - 8 x + 12$ .

How did it go?


Since a is positive, the parabola opens upward.
The quadratic equation has a minimum.
Find the axis of symmetry.		The axis of symmetry is $x = −1$ .
The vertex is on the line $x = −1 .$
Find y when $x = −1 .$		The vertex is $(−1, −9)$ .
Since the parabola has a minimum, the y-coordinate of the vertex is the minimum y-value of the quadratic equation.
The minimum value of the quadratic is $−9$ and it occurs when $x = −1$ .
Show the graph to verify the result.


Since a is positive, the parabola opens upward.
The quadratic equation has a minimum.
Find the axis of symmetry.		The axis of symmetry is $x = −1$ .
The vertex is on the line $x = −1 .$
Find y when $x = −1 .$		The vertex is $(−1, −9)$ .
Since the parabola has a minimum, the y-coordinate of the vertex is the minimum y-value of the quadratic equation.
The minimum value of the quadratic is $−9$ and it occurs when $x = −1$ .
Show the graph to verify the result.