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Learning Module 05B Quadratic Equations & Quadratic Functions

Quadratic Functions

Learning Objectives

In this section, you will:

  • Recognize characteristics of parabolas.
  • Understand how the graph of a parabola is related to its quadratic function.
  • Determine a quadratic function’s minimum or maximum value.
  • Solve problems involving a quadratic function’s minimum or maximum value.

Satellite dishes.

Curved antennas, such as the ones shown above are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.

Recognizing Characteristics of Parabolas

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.

Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.

The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of x at which y=0 .

Identifying the Characteristics of a Parabola

Determine the vertex, axis of symmetry, zeros, and y \text{-}intercept of the parabola shown.

Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).

Show Solution

The vertex is the turning point of the graph. We can see that the vertex is at \left(3,1\right). Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is x=3. This parabola does not cross the xtext{-}axis, so it has no zeros. It crosses the ytext{-}axis at \left(0,7\right) so this is the y-intercept.

Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions

The general form of a quadratic function presents the function in the form

f\left(x\right)=a{x}^{2}+bx+c

where a,b, and c are real numbers and a \ne 0. If a>0, the parabola opens upward. If a<0, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.

The axis of symmetry is defined by x=-\dfrac{b}{2a}. If we use the quadratic formula, x=\dfrac{-b±\sqrt{{b}^{2}-4ac}}{2a}, to solve a{x}^{2}+bx+c=0 for the x\text{-}intercepts, or zeros, we find the value of x halfway between them is always x=-\dfrac{b}{2a}, the equation for the axis of symmetry.

The graph below of the quadratic function written in general form as y={x}^{2}+4x+3. In this form, a=1,b=4, and c=3. Because a>0, the parabola opens upward. The axis of symmetry is x=-\dfrac{4}{2\left(1\right)}=-2. This also makes sense because we can see from the graph that the vertical line x=-2 divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, \left(-2,-1\right). The x\text{-}intercepts, those points where the parabola crosses the x\text{-}axis, occur at \left(-3,0\right) and \left(-1,0\right) .

Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.

The standard form of a quadratic function presents the function in the form

f\left(x\right)=a{\left(x-h\right)}^{2}+k

where \left(h, k\right) is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.

As with the general form, if a>0, the parabola opens upward and the vertex is a minimum. If a<0, the parabola opens downward, and the vertex is a maximum. The graph below represents the graph of the quadratic function written in standard form as y=-3{\left(x+2\right)}^{2}+4. Since x-h=x+2 in this example, h=-2. In this form, a=-3,h=-2, and k=4. Because a<0, the parabola opens downward. The vertex is at \left(-2, 4\right) .

Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.

The standard form is useful for determining how the graph is transformed from the graph of y={x}^{2}. The graph of this basic function.

Graph of y=x^2.

If k>0, the graph shifts upward, whereas if k<0, the graph shifts downward. In the graph k>0, so the graph is shifted 4 units upward. If h>0, the graph shifts toward the right and if h<0, the graph shifts to the left. In the graph, h<0, so the graph is shifted 2 units to the left. The magnitude of a indicates the stretch of the graph. If|a|>1 the point associated with a particular x\text{-}value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if |a|<1, the point associated with a particular x\text{-}value shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In the graph, |a|>1, so the graph becomes narrower.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.

    \begin{align*} a(x-h)^{2} + k &= ax^{2} + bx + c \\ ax^{2} - 2ahx + \big(a h^{2} + k\big) &= ax^{2} + bx + c \end{align*}

For the linear terms to be equal, the coefficients must be equal.

-2ah=b, \text{ so }h=-\dfrac{b}{2a}

This is the axis of symmetry we defined earlier. Setting the constant terms equal:

    \begin{align*} a(x-h)^{2} + k &= ax^{2} + bx + c \\ ax^{2} - 2ahx + (ah^{2} + k) &= ax^{2} + bx + c \end{align*}

In practice, though, it is usually easier to remember that k is the output value of the function when the input is h, so f\left(h\right)=k .

Forms of Quadratic Functions

A quadratic function is a polynomial function of degree two. The graph of a quadratic function is a parabola.

The general form of a quadratic function is f\left(x\right)=a{x}^{2}+bx+c where a,b, and c are real numbers and a \ne 0 .

The standard form of a quadratic function is f\left(x\right)=a{\left(x-h\right)}^{2}+k where a \ne 0 .

The vertex \left(h,k\right) is located at

h=-\dfrac{b}{2a} k=f\left(h\right)=f\left(\dfrac{-b}{2a}\right)

How To

Given a graph of a quadratic function, write the equation of the function in general form.

  1. Identify the horizontal shift of the parabola; this value is h. Identify the vertical shift of the parabola; this value is k .
  2. Substitute the values of the horizontal and vertical shift for h and k. in the function f\left(x\right)=a{\left(x-h\right)}^{2}+k .
  3. Substitute the values of any point, other than the vertex, on the graph of the parabola for x and f\left(x\right) .
  4. Solve for the stretch factor, |a| .
  5. Expand and simplify to write in general form.

Writing the Equation of a Quadratic Function from the Graph

Write an equation for the quadratic function g as a transformation of f\left(x\right)={x}^{2}, and then expand the formula, and simplify terms to write the equation in general form.

Graph of a parabola with its vertex at (-2, -3).

Show Solution

We can see the graph of g is the graph of f\left(x\right)={x}^{2} shifted to the left 2 and down 3, giving a formula in the form g\left(x\right)=a{\left(x-\left(-2\right)\right)}^{2}-3=a{\left(x+2\right)}^{2}-3 .

Substituting the coordinates of a point on the curve, such as \left(0,-1\right), we can solve for the stretch factor.

    \begin{align*} -1 &= a(0+2)^{2} - 3 \\ 2 &= 4a \\ a &= \tfrac{1}{2} \end{align*}

In standard form, the algebraic model for this graph is \left(g\right)x=\dfrac{1}{2}{\left(x+2\right)}^{2}-3 .

To write this in general polynomial form, we can expand the formula and simplify terms.

    \begin{align*} g(x) &= \frac{1}{2}(x+2)^{2} - 3 \\ &= \frac{1}{2}(x+2)(x+2) - 3 \\ &= \frac{1}{2}(x^{2} + 4x + 4) - 3 \\ &= \frac{1}{2}x^{2} + 2x + 2 - 3 \\ &= \frac{1}{2}x^{2} + 2x - 1 \end{align*}

Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.

 

Try It

A coordinate grid has been superimposed over the quadratic path of a basketball. Find an equation for the path of the ball. Does the shooter make the basket?

Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.

Show Solution

The path passes through the origin and has vertex at \left(-4, 7\right), so \left(h\right)x=-\dfrac{7}{16}{\left(x+4\right)}^{2}+7. To make the shot, h\left(-7.5\right) would need to be about 4 but h\left(-7.5\right) \approx 1.64; he doesn’t make it.

How To

Given a quadratic function in general form, find the vertex of the parabola.

  1. Identify a, b, \text{and} c .
  2. Find h, the x-coordinate of the vertex, by substituting a and b into h=-\dfrac{b}{2a} .
  3. Find k, the y-coordinate of the vertex, by evaluating k=f\left(h\right)=f\left(-\dfrac{b}{2a}\right) .

Finding the Vertex of a Quadratic Function

Find the vertex of the quadratic function f\left(x\right)=2{x}^{2}-6x+7. Rewrite the quadratic in standard form (vertex form).

Show Solution

    \begin{align*} \text{The horizontal coordinate of the vertex will be at:} \quad h &= -\frac{b}{2a} \\ &= -\frac{-6}{2(2)} \\ &= \frac{6}{4} \\ &= \frac{3}{2} \\ \text{The vertical coordinate of the vertex will be at:} \quad k &= f(h) \\ &= f\!\left(\frac{3}{2}\right) \\ &= 2\left(\frac{3}{2}\right)^{2} - 6\left(\frac{3}{2}\right) + 7 \\ &= \frac{5}{2} \end{align*}

Rewriting into standard form, the stretch factor will be the same as the a in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the “a” from the general form.

    \begin{align*} f(x) &= ax^{2} + bx + c \\ f(x) &= 2x^{2} - 6x + 7 \end{align*}

The standard form of a quadratic function prior to writing the function then becomes the following:

f\left(x\right)=2{\left(x-\dfrac{3}{2}\right)}^{2}+\dfrac{5}{2}

Analysis

One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, k and where it occurs, x .

Try It

Given the equation g\left(x\right)=13+{x}^{2}-6x write the equation in general form and then in standard form.

Show Solution

g\left(x\right)={x}^{2}-6x+13 in general form; g\left(x\right)={\left(x-3\right)}^{2}+4 in standard form

Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.

Domain and Range of a Quadratic Function

The domain of any quadratic function is all real numbers unless the context of the function presents some restrictions.

The range of a quadratic function written in general form f\left(x\right)=a{x}^{2}+bx+c with a positive a value is f\left(x\right) \ge f\left(-\dfrac{b}{2a}\right), or left[f\left(-\dfrac{b}{2a}\right),infty \right); the range of a quadratic function written in general form with a negative a value is f\left(x\right)le f\left(-\dfrac{b}{2a}\right), or \left(-infty ,f\left(-\dfrac{b}{2a}\right)right] .

The range of a quadratic function written in standard form f\left(x\right)=a{\left(x-h\right)}^{2}+k with a positive a value is f\left(x\right) \ge k; the range of a quadratic function written in standard form with a negative a value is f\left(x\right)le k .

How To

Given a quadratic function, find the domain and range.

  1. Identify the domain of any quadratic function as all real numbers.
  2. Determine whether a is positive or negative. If a is positive, the parabola has a minimum. If a is negative, the parabola has a maximum.
  3. Determine the maximum or minimum value of the parabola, k .
  4. If the parabola has a minimum, the range is given by f\left(x\right) \ge k, or \left[k,infty \right). If the parabola has a maximum, the range is given by f\left(x\right) \le k, or \left(-infty ,k \right] .

Finding the Domain and Range of a Quadratic Function

Find the domain and range of f\left(x\right)=-5{x}^{2}+9x-1 .

Show Solution

As with any quadratic function, the domain is all real numbers.

Because a is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the xtext{-}value of the vertex.

    \begin{align*} h &= -\frac{b}{2a} \\ &= -\frac{9}{2(-5)} \\ &= \frac{9}{10} \end{align*}

The maximum value is given by f\left(h\right) .

    \begin{align*} f\left(\frac{9}{10}\right) &= -5\left(\frac{9}{10}\right)^{2} + 9\left(\frac{9}{10}\right) - 1 \\ &= \frac{61}{20} \end{align*}

The range is f\left(x\right) \le \dfrac{61}{20}, or \left(-infty ,\dfrac{61}{20} \right] .

Determining the Maximum and Minimum Values of Quadratic Functions

The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola. We can see the maximum and minimum values.

Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).

There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.

Finding the Maximum Value of a Quadratic Function

A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.

  1. Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length L .
  2. What dimensions should she make her garden to maximize the enclosed area?
Show Solution

Let’s use a diagram to record the given information. It is also helpful to introduce a temporary variable, W, to represent the width of the garden and the length of the fence section parallel to the backyard fence.

Diagram of the garden and the backyard.

  1. We know we have only 80 feet of fence available, and L+W+L=80, or more simply, 2L+W=80. This allows us to represent the width, W, in terms of L .
    W=80-2L

    Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so

        \begin{align*} A &= LW = L(80 - 2L) \\ A(L) &= 80L - 2L^{2} \end{align*}

    This formula represents the area of the fence in terms of the variable length L. The function, written in general form, is

    A\left(L\right)=-2{L}^{2}+80L .
  2. The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since a is the coefficient of the squared term, a=-2,b=80, and c=0 .

To find the vertex:

    \begin{align*} h &= -\frac{b}{2a} & \quad k &= A(20) \\ &= -\frac{80}{2(-2)} & &= 80(20) - 2(20)^{2} \\ &= 20 & &= 800 \end{align*}

The maximum value of the function is an area of 800 square feet, which occurs when L=20 feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.

Analysis

This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function.

Graph of the parabolic function A(L)=-2L^2+80L, which the x-axis is labeled Length (L) and the y-axis is labeled Area (A). The vertex is at (20, 800).

Applying the Vertex and x-Intercepts of a Parabola

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation H\left(t\right)=-16{t}^{2}+80t+40 .

  1. When does the ball reach the maximum height?
  2. What is the maximum height of the ball?
  3. When does the ball hit the ground?
Show Solution

 

  1. The ball reaches the maximum height at the vertex of the parabola.

        \begin{align*} h &= -\frac{80}{2(-16)} \\ &= \frac{80}{32} \\ &= \frac{5}{2} \\ &= 2.5 \end{align*}

    The ball reaches a maximum height after 2.5 seconds.

  2. To find the maximum height, find the y\text{-}coordinate of the vertex of the parabola.

        \begin{align*} k &= H\left(-\frac{b}{2a}\right) \\ &= H(2.5) \\ &= -16(2.5)^{2} + 80(2.5) + 40 \\ &= 140 \end{align*}

    The ball reaches a maximum height of 140 feet.

  3. To find when the ball hits the ground, we need to determine when the height is zero, H\left(t\right)=0 .We use the quadratic formula.

        \begin{align*} t &= \frac{-80 \pm \sqrt{80^{2} - 4(-16)(40)}}{2(-16)} \\ &= \frac{-80 \pm \sqrt{8960}}{-32} \end{align*}

    Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.

        \begin{align*} t &= \frac{-80 - \sqrt{8960}}{-32} \approx 5.458 \quad \text{or} \quad t = \frac{-80 + \sqrt{8960}}{-32} \approx -0.458 \end{align*}

    The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds.

    Graph of a negative parabola where x goes from -1 to 6.

    Note that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph.

Try It

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation H\left(t\right)=-16{t}^{2}+96t+112 .

  1. When does the rock reach the maximum height?
  2. What is the maximum height of the rock?
  3. When does the rock hit the ocean?
Show Solution

3 seconds256 feet7 seconds

Access these online resources for additional instruction and practice with quadratic equations.

Key Concepts

  • A polynomial function of degree two is called a quadratic function.
  • The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
  • The axis of symmetry is the vertical line passing through the vertex. The zeros, or x \text{-}intercepts, are the points at which the parabola crosses the x \text{-}axis. The y \text{-}intercept is the point at which the parabola crosses the y \text{-}axis.
  • Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
  • The vertex can be found from an equation representing a quadratic function.
  • The domain of a quadratic function is all real numbers. The range varies with the function.
  • A quadratic function’s minimum or maximum value is given by the y \text{-}value of the vertex.
  • 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.

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