6.5 Problem Solving and Quadratic Functions

Learning Outcomes

  • Solve maximum and minimum applications by finding the vertex of a given quadratic function.
  • Solve applications modeled by quadratic functions.

In this section, we will use polynomial functions to answer questions about the parabolic motion of a projectile. The real mathematical model for the path of a rocket or a police GPS projectile may have different coefficients or more variables, but the concept remains the same. We will also learn to interpret the meaning of the variables in a polynomial function that models projectile motion.

Finding the x– and y-Intercepts of a Quadratic Function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the y-intercept of a quadratic by evaluating the function at an input of zero, and we find the x-intercepts at locations where the output is zero. Notice that the number of x-intercepts can vary depending upon the location of the graph.

Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one –intercept, and the third parabola is of two x-intercepts.

Figure 13. Number of x-intercepts of a parabola

How To: Given a quadratic function [latex]f\left(x\right)[/latex], find the y– and x-intercepts.

  1. Evaluate [latex]f\left(0\right)[/latex] to find the y-intercept.
  2. Solve the quadratic equation [latex]f\left(x\right)=0[/latex] to find the x-intercepts.

Example 7: Finding the y– and x-Intercepts of a Parabola

Find the y– and x-intercepts of the quadratic [latex]f\left(x\right)=3{x}^{2}+5x - 2[/latex].

  • The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.
  • The vertex and the intercepts can be identified and interpreted to solve real-world problems.

Solve problems involving a quadratic function’s minimum or maximum value

In Example 7, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

How To: Given a quadratic function, find the x-intercepts by rewriting in standard form.

  1. Substitute a and b into [latex]h=-\frac{b}{2a}[/latex].
  2. Substitute x = h into the general form of the quadratic function to find k.
  3. Rewrite the quadratic in standard form using h and k.
  4. Solve for when the output of the function will be zero to find the x-intercepts.

Example 8: Finding the x-Intercepts of a Parabola

Find the x-intercepts of the quadratic function [latex]f\left(x\right)=2{x}^{2}+4x - 4[/latex].

try it