Finding the Minimum or Maximum of Quadratic Functions | Mathematics
Now that you’re skilled in graphing quadratic functions, let’s take a look at another high emphasis topic for the TASC Test Assessing Secondary Completion™ Mathematics subtest: finding the minimum or maximum of a quadratic function.
We know that the parabola in the graph below is positive because of its “smile shape.”
Positive parabolas don’t have a definite maximum value since the two sides of the parabola extend upward indefinitely.
Positive parabolas have a definite minimum value. The lowest point, or minimum value, is highlighted in yellow. You may also remember this point as the parabola’s vertex.
To find the minimum value, find where the vertex aligns on the y-axis. In this case, the vertex is (2, -4) and the minimum value is -4.
We know that the parabola in the graph below is negative because of its “rainbow shape.”
Negative parabolas don’t have a definite minimum value since the two sides of the parabola extend downward indefinitely.
Negative parabolas have a definite maximum value. The highest point, or maximum value, is highlighted in yellow. Again, this is also the parabola’s vertex.
Find where the vertex aligns on the y-axis; this is the parabola’s maximum value. In this case, the vertex is (1, 4) and the maximum value is 4.
Steps to Find a Quadratic Function’s Minimum or Maximum
Find the minimum or maximum value of f(x)= -x2 - 4x – 5.
- Using the standard quadratic form variables (a, b, c), you can find the x-coordinate through the following equation: (Remember, the standard quadratic form is: f(x) = ax2 + bx + c.)
- Next, plug the x-coordinate of -2 into the function f(x)= -x2 - 4x – 5 to find the y-coordinate.
f(-2)= -(-2)2 – 4(-2) – 5
f(-2)= -4 – (-8) – 5
f(-2)= -1 = y-coordinate
- We know the vertex now to be (-2, -1).
- Since “-x2” is negative, we know the parabola is a rainbow shape.
- This parabola has a maximum value of -1, and no minimum value. Remember, this is the value the vertex lines up to on the y-axis.
- You can draw an estimated parabola knowing the maximum value of -1, the vertex of (-2, -1), and the shape of the parabola: