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MAT 153: College Algebra and Trigonometry

Section 1.4 Quadratic Functions

In this section we introduce quadratic functions. The graph of a quadratic function is a parabola. In addition to the intercepts, we will be interested in finding the vertex of the parabola, the highest or lowest point on the parabola.
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Goals:
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  • Q: Be able to solve a quadratic equation.
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  • Q: Be able to determine the vertex and equation of a quadratic function given its graph.
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  • F: Be able to give the solution to an inequality or set of inequalities using proper mathematical notation.
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  • F: Be able to determine the domain or range of a function given as an equation or a graph.
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Definition 1.4.1.

A quadratic function is a function that can be written as
\begin{equation*} f(x)=ax^2+bx+c \end{equation*}
where \(a \neq 0\text{.}\) The shape of the graph of a quadratic function is a parabola.
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The minimum or maximum value of a quadratic function is at the vertex of its associated parabola.
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A quadratic function can also be written in the form \(f(x)=a(x-h)^2+k\text{,}\) where \(a \neq 0\) and \(h\) and \(k\) are real numbers.
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Investigation 1.4.1.

In Desmos, graph \(f(x)\) and create sliders for \(a\text{,}\) \(h\) and \(k\text{.}\) What role do each of these constants play? Include sketches and verbal descriptions to help explain the role of each constant.
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Remark 1.4.2.

Sometimes we will have the graph of the quadratic function. Other times we have the equation for the quadratic function, but we may have to do some work to put it the form in Definition 1.4.1. As we’ve seen with linear functions, the names of the variables don’t matter. Instead of \(x\) you might see \(t\) or another variable, and instead of \(f\) the function may be named with a different letter.
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Solving Quadratic Equations
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Quadratic equations can be solved in several ways: by factoring, by using the quadratic formula, by completing the square or by using a graph. We will first consider solving quadratic equations by factoring, and see how this compares to solving quadratic equations using a graph.
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Definition 1.4.3.

To solve quadratic equations by factoring, we will use the zero-product property, which says that if \(ab=0\text{,}\) then \(a=0\) or \(b=0\text{.}\)
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Problem 1.4.4.

Solve each equation below by factoring.
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  1. \(\displaystyle x^2 - 5x - 6 = 0\)
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  2. \(\displaystyle a^2 + 2 = -3a\)
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  3. \(\displaystyle r^2 - 16 = 0\)
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  4. \(\displaystyle 12x^2 + 8x - 15 = 0\)
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  5. \(\displaystyle v^2 = 5v\)
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  6. \(\displaystyle -4x^2 - 20x + 24 = 0\)
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Some equations can be solved by factoring, even if they are not quadratic equations of the form \(ax^2+bx+c=0\text{.}\)
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Example 1.4.5.

Solve by factoring.
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  1. \(\displaystyle n^4 - 4n^2 + 3 = 0\)
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  2. \(\displaystyle 2f^4 = 6f^3 + 8f^2\)
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  3. \(\displaystyle 14r^3 + 7r^2 = 7r\)
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  4. \(\displaystyle x^4 - 2x^2 + 1 = 0\)
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Not every quadratic equation can be solved by factoring. One method of solving any quadratic equation is using the quadratic formula.
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Definition 1.4.6.

The quadratic formula is
\begin{equation*} x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{equation*}
where \(a\text{,}\) \(b\text{,}\) and \(c\) are coefficients of the quadratic equation \(ax^2+bx+c=0\text{.}\)
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Example 1.4.7.

Solve \(2x^2-x-4=3\)
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Problem 1.4.8.

Find the \(x-\)intercepts of each quadratic function. Use Desmos to sketch a graph of the function and label the \(x-\)intercepts.
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  1. \(\displaystyle f(x)=-3x^2-x+3\)
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  2. \(\displaystyle g(x)=2x^2-4x+5\)
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  3. \(\displaystyle k(x)=4x^2-12x+9\)
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  4. From your answers to Item 1 through Item 3 above, explain how you can tell if a quadratic equation has no solutions, one solution or two solutions.
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Example 1.4.9.

Refer to the graph of \(k\) in Figure Figure 1.4.10.
A graph of a quadratic function labeled k of x. The parabola opens upward and crosses the x-axis near x equals 2 and x equals 6. The vertex occurs near x equals 4 with a minimum value around negative 10, and the graph is symmetric about the vertical line x equals 4.
Figure 1.4.10. Graph of the function k(x).
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  1. What are the coordinates of the vertex of the graph of \(k\text{?}\)
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  2. Write a function equation for \(k(x)\text{.}\)
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  3. Solve \(k(x)=15\) for \(x\text{.}\)
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  4. Solve the inequality \(k(x) \leq 15\text{.}\)
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  5. What is the range of \(k\) on the domain of all real numbers? What is the range of \(k\) on the domain \(0 \leq x \leq 6\text{?}\)
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Problem 1.4.11.

Refer to the graph of \(q\) in Figure 1.4.12.
A graph of a quadratic function labeled q of x. The parabola opens downward and has a maximum near x equals negative 3 with a value just below zero. The graph decreases on both sides of the vertex and extends downward to large negative values.
Figure 1.4.12. Graph of the function q(x).
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  1. Evaluate \(q(-2)\text{.}\)
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  2. What are the coordinates of the vertex of \(q(x)\text{?}\)
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  3. Write a function equation for \(q(x)\text{.}\)
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  4. Use your equation from Item 3 to find \(q(-5)\text{.}\) Verify the point on your graph.
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  5. Solve the inequality \(q(x) \leq -6\text{.}\)
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  6. Solve the inequality \(q(x) > 2x - 12\text{.}\)
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  7. What is the range of \(q\) on the domain of all real numbers?
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Example 1.4.13.

Let \(f(x)=-x^2+2x+3\)
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  1. Find the \(x-\) and \(y-\)intercepts of the graph of \(f\text{.}\)
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  2. Find the vertex of the graph of \(f\text{.}\)
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  3. Use Item 1 and Item 2 to graph \(f\text{.}\) Use Desmos to check your answer.
    A blank coordinate grid showing the horizontal x-axis and vertical f(x)-axis intersecting at the origin. The x-axis extends in both directions and is labeled x, and the vertical axis is labeled f(x) with values increasing upward.
    Figure 1.4.14. Coordinate axes for the graph of f(x).
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  4. Find all the values of \(x\) where \(f(x)=3\text{.}\)
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  5. Solve \(f(x) \leq 3\text{.}\)
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  6. Solve \(f(x)=1\) for \(x\text{.}\) Give your answer as an exact value and a decimal approximation.
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Problem 1.4.15.

Let \(g(x)=-2x^2+4x+1\text{.}\)
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  1. Write \(g\) in vertex form.
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  2. Graph \(g\) on the axes below.
    A blank coordinate grid with the horizontal x-axis and vertical g(x)-axis intersecting at the origin. The x-axis is labeled x and extends to positive and negative values, and the vertical axis is labeled g(x) with values increasing upward.
    Figure 1.4.16. Coordinate axes for the graph of g(x).
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  3. Solve \(g(x)=0\text{.}\)
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  4. Solve \(g(x)=-4\text{.}\)
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  5. What is the domain of \(g\text{?}\) What is the range of \(g\text{?}\)
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  6. What is the maximum value of \(g\text{?}\) For what value of \(x\) does this maximum occur?
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Problem 1.4.17.

Refer to the function \(r(t)=\frac{1}{2}t^2 - \frac{9}{2}t + 4\text{.}\)
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  1. Graph \(r\) on the domain \(-2 \leq t \leq 14\text{.}\) Be sure to label the vertex and any intercept(s).
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  2. What is the range of \(r\) corresponding to the domain \(-2 \leq t \leq 14\text{?}\)
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  3. What is the range of \(r\) on the domain of all real numbers?
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