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Section 11.3 Using Technology to Explore Functions

Graphing technology allows us to explore the properties of functions more deeply than we can with only pencil and paper. It can quickly create a table of values, and quickly plot the graph of a function. Such technology can also evaluate functions, solve equations with functions, find maximum and minimum values, and explore other key features.
There are many graphing technologies currently available, including (but not limited to) physical (hand-held) graphing calculators, Desmos
 1 
www.desmos.com
, GeoGebra
 2 
www.geogebra.org
, Sage
 3 
www.sagemath.org
, and WolframAlpha
 4 
www.wolframalpha.com
.
This section will focus on how technology can be used to explore functions and their key features. Although the choice of particular graphing technology varies by each school and curriculum, the main ways in which technology is used to explore functions is the same and can be done with each of the technologies above.
Figure 11.3.1. Alternative Video Lesson

Subsection 11.3.1 Finding an Appropriate Window

With a simple linear equation like \(y=2x+5\text{,}\) most graphing technologies will show this graph in a good window by default. A common default window goes from \(x=-10\) to \(x=10\) and \(y=-10\) to \(y=10\text{.}\)
What if we wanted to graph something with a much larger magnitude though, such as \(y=2000x+5000\text{?}\) If we tried to view this for \(x=-10\) to \(x=10\) and \(y=-10\) to \(y=10\text{,}\) the function would appear as an almost vertical line since it has such a steep slope.
Using technology, we will create a table of values for this function as shown in Figure 11.3.2.(a). Then we will set the \(x\)-values for which we view the function to go from \(x=-5\) to \(x=5\) and the \(y\)-values from \(y=-20{,}000\) to \(y=20{,}000\text{.}\) The graph is shown in Figure 11.3.2.(b).
\(x\) \(y=2000x+5000\)
\(-5\) \(-5000\)
\(-4\) \(-3000\)
\(-3\) \(-1000\)
\(-2\) \(1000\)
\(-1\) \(3000\)
\(0\) \(5000\)
\(1\) \(7000\)
\(2\) \(9000\)
\(3\) \(11000\)
\(4\) \(13000\)
\(5\) \(15000\)
(a) A table of values
(b) Graphed with an appropriate window
Figure 11.3.2. Creating a table of values to determine an appropriate graphing window
Now let’s practice finding an appropriate viewing window with a less familiar function.

Example 11.3.3.

Find an appropriate window for \(q(x)=\frac{x^3}{100}-2x+1\text{.}\)
Entering this function into graphing technology, we input q(x)=(x^3)/100-2x+1. A default window will generally give us something like this:
Figure 11.3.4. Function \(q\) graphed in the default window.
We can tell from the lower right corner of Figure 4 that we’re not quite viewing all of the important details of this function. To determine a better window, we could use technology to make a table of values. Another more rudimentary option is to double the viewing constraints for \(x\) and \(y\text{,}\) as shown in Figure 5. Many graphing technologies have the ability to zoom in and out quickly.
Figure 11.3.5. Function \(q\) graphed in an expanded window.

Subsection 11.3.2 Using Technology to Determine Key Features of a Graph

The key features of a graph can be determined using graphing technology. Here, we’ll show how to determine the \(x\)-intercepts, \(y\)-intercepts, and maximum/minimum values using technology.

Example 11.3.6.

Graph the function given by \(p(x)=-1000x^2-100x+40\text{.}\) Determine an appropriate viewing window, and then use graphing technology to determine the following:
  1. Determine the \(x\)-intercepts of the function.
  2. Determine the \(y\)-intercept of the function.
  3. Determine the maximum function value and where it occurs.
Explanation.
To start, we’ll take a quick view of this function in a default window. We can see that we need to zoom in on the \(x\)-values, but we need to zoom out on the \(y\)-values.
From the graph we see that the \(x\)-values might as well run from about \(-0.5\) to \(0.5\text{,}\) so we will look at \(x\)-values in that window in increments of \(0.1\text{,}\) as shown in Table 11.3.8.(a). This table allows us to determine an appropriate viewing window for \(y=p(x)\) which is shown in Figure 11.3.8.(b). The table suggests we should go a little higher than \(40\) on the \(y\)-axis, and it would be OK to go the same distance in the negative direction to keep the \(x\)-axis centered.
Figure 11.3.7. Graph of \(y=p(x)\) in an inappropriate window
\(x\) \(p(x)\)
\(-0.5\) \(-160\)
\(-0.4\) \(-80\)
\(-0.3\) \(-20\)
\(-0.2\) \(20\)
\(-0.1\) \(40\)
\(0\) \(40\)
\(0.1\) \(20\)
\(0.2\) \(-20\)
\(0.3\) \(-80\)
\(0.4\) \(-160\)
\(0.5\) \(-260\)
(a) Function values for \(y=p(x)\)
(b) Graph of \(y=p(x)\) in an appropriate window showing key features
Figure 11.3.8. Creating a table of values to determine an appropriate graphing window
We can now use Figure 11.3.8.(b) to determine the \(x\)-intercepts, the \(y\)-intercept, and the maximum function value.
  1. To determine the \(x\)-intercepts, we will find the points where \(y\) is zero. These are about \((-0.2562,0)\) and \((0.1562,0)\text{.}\)
  2. To determine the \(y\)-intercept, we need the point where \(x\) is zero. This point is \((0,40)\text{.}\)
  3. The highest point on the graph is the vertex, which is about \((-0.05,42.5)\text{.}\) So the maximum function value is \(42.5\) and occurs at \(-0.05\text{.}\)
  4. We can see that the function is defined for all \(x\)-values, so the domain is \((-\infty,\infty)\text{.}\) The maximum function value is \(42.5\text{,}\) and there is no minimum function value. Thus the range is \((-\infty,42.5]\text{.}\)

Example 11.3.9. Graphing Technology Limitations.

If we use graphing technology to graph the function \(g\) where \(g(x) = 0.0002x^2 + 0.00146x + 0.00266\text{,}\) we may be mislead by the way values are rounded. Without technology, we know that this function is a quadratic function and therefore has at most two \(x\)-intercepts and has a vertex that will determine the minimum function value. However, using technology we could obtain a graph with the following key points:
Figure 11.3.10. Misleading graph
This looks like there are three \(x\)-intercepts, which we know is not possible for a quadratic function. We can evaluate \(g\) at \(x=-3.65\) and determine that \(g(-3.65)= -0.0000045\text{,}\) which is approximately zero when rounded. So the true vertex of this function is \((-3.65,-0.0000045)\text{,}\) and the minimum value of this function is \(-0.0000045\) (not zero).
Every graphing tool generally has some type of limitation like this one, and it’s good to be aware that these limitations exist.

Subsection 11.3.3 Solving Equations Graphically Using Technology

To algebraically solve an equation like \(h(x)=v(x)\) for
\begin{align*} h(x)\amp=-0.01(x-90)(x+20)\amp\amp\text{and}\amp v(x)\amp=-0.04(x-10)(x-80)\text{,} \end{align*}
we’d start by setting up
\begin{equation*} -0.01(x-90)(x+20)=-0.04(x-10)(x-80) \end{equation*}
To solve this, we’d then simplify each side of the equation, set it equal to zero, and finally use the quadratic formula like we did in Section 7.2.
An alternative is to graphically solve this equation, which is done by graphing
\begin{align*} y\amp=-0.01(x-90)(x+20)\amp\amp\text{and}\amp y\amp=-0.04(x-10)(x-80)\text{.} \end{align*}
Figure 11.3.11. Points of intersection for \(h(x)=v(x)\)
The points of intersection, \((22.46,28.677)\) and \((74.207,14.878)\text{,}\) show where these functions are equal. This means that the \(x\)-values give the solutions to the equation \(-0.01(x-90)(x+20)=-0.04(x-10)(x-80)\text{.}\) So the solutions are approximately \(22.46\) and \(74.207\text{,}\) and the solution set is approximately \(\{22.46, 74.207\}\text{.}\)
Similarly, to graphically solve an equation like \(h(x)=25\) for
\begin{equation*} h(x)=-0.01(x-90)(x+20)\text{,} \end{equation*}
we can graph
\begin{equation*} y=-0.01(x-90)(x+20)\qquad\text{and}\qquad y=25 \end{equation*}
Figure 11.3.12. Points of intersection for \(h(x)=25\)
The points of intersection are \((12.807,25)\) and \((57.913,25)\text{,}\) which tells us that the solutions to \(h(x)=25\) are approximately \(12.807\) and \(57.913\text{.}\) The solution set is approximately \(\{12.807, 57.913\}\text{.}\)

Example 11.3.13.

Use graphing technology to solve the following inequalities:
  1. \(\displaystyle -20t^2-70t+300 \geq -5t+300\)
  2. \(\displaystyle -20t^2-70t+300 \lt -5t+300\)
Explanation.
To solve these inequalities graphically, we will start by graphing the equations \(y=-20t^2-70t+300\) and \(y=-5t+300\) and determining the points of intersection:
Figure 11.3.14. Points of intersection for \(y=-20t^2-70t+300\) and \(y=-5t+300\)
  1. To solve \(-20t^2-70t+300 \geq -5t+300\text{,}\) we need to determine where the \(y\)-values of the graph of \(y=-20t^2-70t+300\) are greater than the \(y\)-values of the graph of \(y=-5t+300\) in addition to the values where the \(y\)-values are equal. This region is highlighted in Figure 15.
    Figure 11.3.15.
    We can see that this region includes all values of \(t\) between, and including, \(t=-3.25\) and \(t=0\text{.}\) So the solutions to this inequality include all values of \(t\) for which \(-3.25\le t \le 0\text{.}\) We can write this solution set in interval notation as \([-3.25,0]\) or in set-builder notation as \(\{t\mid -3.25 \leq t \leq 0\}\text{.}\)
  2. To now solve \(-20t^2-70t+300 \lt -5t+300\text{,}\) we will need to determine where the \(y\)-values of the graph of \(y=-20t^2-70t+300\) are less than the \(y\)-values of the graph of \(y=-5t+300\text{.}\) This region is highlighted in Figure 16.
    Figure 11.3.16.
    We can see that \(-20t^2-70t+300 \lt -5t+300\) for all values of \(t\) where \(t\lt -3.25\) or \(t\gt 0\text{.}\) We can write this solution set in interval notation as \((-\infty,-3.25)\cup(0,\infty)\) or in set-builder notation as \(\{t\mid t\lt -3.25 \text{ or } t\gt 0\}\text{.}\)

Reading Questions 11.3.4 Reading Questions

1.

If you use technology to create a graph of a function, you will have to choose a good to capture all of its features.

2.

Describe the process to graphically solve an equation in the form \(f(x)=g(x)\text{.}\)

Exercises 11.3.5 Exercises

Using Technology to Create a Table of Function Values.

1.
Use technology to make a table of values for the function.
\(g(x)={x^{2}-7x-2}\)
\(x\) \(g(x)\)
2.
Use technology to make a table of values for the function.
\(h(x)={-2x^{2}+9x-5}\)
\(x\) \(h(x)\)
3.
Use technology to make a table of values for the function.
\(F(x)={3.45x^{2}-150x+47}\)
\(x\) \(F(x)\)
4.
Use technology to make a table of values for the function.
\(G(x)={0.7x^{2}+30x-9}\)
\(x\) \(G(x)\)
5.
Use technology to make a table of values for the function.
\(H(x)={-7x^{3}+200x-64}\)
\(x\) \(H(x)\)
6.
Use technology to make a table of values for the function.
\(H(x)={10x^{3}-40x+81}\)
\(x\) \(H(x)\)

Determining Appropriate Windows.

7.
Let \(f(x)={9464x+8319}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
8.
Let \(f(x)={-1461x-3072}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
9.
Let \(f(x)={170x-650}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
10.
Let \(f(x)={-279x+73}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
11.
Let \(f(x)={-389x^{2}+486x+3465}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
12.
Let \(f(x)={499x^{2}-898x-3336}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
13.
Let \(f(x)={-0.00034x^{2}+0.0022x+0.35}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .
14.
Let \(f(x)={-0.00039x^{2}+0.0039x-0.34}\text{.}\) Choose an appropriate window for graphing \(f\) that shows its key features.
The \(x\)-interval could be and the \(y\)-interval could be .

Finding Points of Intersection.

15.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={\left(-352+13x\right)\mathopen{}\left(-92+11x\right)}\) and \(y={-9000}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
16.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={\left(65+18x\right)\mathopen{}\left(-369-4x\right)}\) and \(y={-9000}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
17.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={-9x^{3}-3x^{2}+9x}\) and \(y={7x-5}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
18.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={-6x^{3}+2x^{2}-2x}\) and \(y={x-5}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
19.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={-0.5\mathopen{}\left(3x^{2}+6\right)}\) and \(y={0.05\mathopen{}\left(4x-4\right)}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
20.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={-0.5\mathopen{}\left(4x^{2}-5\right)}\) and \(y={-0.86\mathopen{}\left(x-9\right)}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
21.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={-0.05\mathopen{}\left(x-3\right)^{2}+2.7}\) and \(y={x-1}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.
22.
Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
\(y={0.4\mathopen{}\left(x+3\right)^{2}-7.8}\) and \(y={-0.15x-2}\) intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
.

Using Technology to Find Key Features of a Graph.

23.
For the function \(j\) defined by
\begin{equation*} j(x)=-\frac{2}{5}(x-3)^2+6 \text{,} \end{equation*}
use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
24.
For the function \(k\) defined by
\begin{equation*} k(x)=2(x+1)^2+10 \text{,} \end{equation*}
use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
25.
For the function \(L\) defined by
\begin{equation*} L(x)=3000x^2+10x+4 \text{,} \end{equation*}
use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
26.
For the function \(M\) defined by
\begin{equation*} M(x)=-(300x-2950)^2 \text{,} \end{equation*}
use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
27.
For the function \(N\) defined by
\begin{equation*} N(x)=(300x-1.05)^2 \text{,} \end{equation*}
use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
28.
For the function \(B\) defined by
\begin{equation*} B(x) = x^2 - 0.05 x + 0.0006 \text{,} \end{equation*}
use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.

Solving Equations Graphically Using Technology.

29.
Let \(s(x)=\frac{1}{5}x^2-2x+10\) and \(t(x)=-x+40\text{.}\) Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve \(s(x)=t(x)\text{.}\)
  3. Solve \(s(x)\gt t(x)\text{.}\)
  4. Solve \(s(x)\leq t(x)\text{.}\)
30.
Let \(w(x)=\frac{1}{4}x^2-3x-8\) and \(m(x)=x+12\text{.}\) Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve \(w(x)=m(x)\text{.}\)
  3. Solve \(w(x)\gt m(x)\text{.}\)
  4. Solve \(w(x)\leq m(x)\text{.}\)
31.
Let \(f(x)=4x^2+5x-1\) and \(g(x)=5\text{.}\) Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve \(f(x)=g(x)\text{.}\)
  3. Solve \(f(x)\lt g(x)\text{.}\)
  4. Solve \(f(x)\geq g(x)\text{.}\)
32.
Let \(p(x)=6x^2-3x+4\) and \(k(x)=7\text{.}\) Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve \(p(x)=k(x)\text{.}\)
  3. Solve \(p(x)\lt k(x)\text{.}\)
  4. Solve \(p(x)\geq k(x)\text{.}\)
33.
Let \(q(x)=-4x^2-24x+10\) and \(r(x)=2x+22\text{.}\) Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve \(q(x)=r(x)\text{.}\)
  3. Solve \(q(x)\gt r(x)\text{.}\)
  4. Solve \(q(x)\leq r(x)\text{.}\)
34.
Let \(h(x)=-10x^2-5x+3\) and \(j(x)=-3x-9\text{.}\) Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve \(h(x)=j(x)\text{.}\)
  3. Solve \(h(x)\gt j(x)\text{.}\)
  4. Solve \(h(x)\leq j(x)\text{.}\)
35.
Use graphing technology to solve the equation \(0.4x^2+0.5x-0.2=2.4\text{.}\) Approximate the solution(s) if necessary.
36.
Use graphing technology to solve the equation \(-0.25x^2-2x+1.75=4.75\text{.}\) Approximate the solution(s) if necessary.
37.
Use graphing technology to solve the equation \((200+5x)(100-2x)=15000\text{.}\) Approximate the solution(s) if necessary.
38.
Use graphing technology to solve the equation \(\left(200-5x\right)\left(100+10x\right)=20000\text{.}\) Approximate the solution(s) if necessary.
39.
Use graphing technology to solve the equation \(2x^3-5x+1=-\frac{1}{2}x+1\text{.}\) Approximate the solution(s) if necessary.
40.
Use graphing technology to solve the equation \(-x^3+8x=-4x+16\text{.}\) Approximate the solution(s) if necessary.
41.
Use graphing technology to solve the equation \(-0.05 x^2 - 2.03 x - 19.6=0.05 x^2 + 1.97 x + 19.4\text{.}\) Approximate the solution(s) if necessary.
42.
Use graphing technology to solve the equation \(-0.02 x^2 + 1.97 x - 51.5=0.05\left(x-50\right)^2-0.03\left(x-50\right)\text{.}\) Approximate the solution(s) if necessary.
43.
Use graphing technology to solve the equation \(-200x^2+60x-55=-20x-40\text{.}\) Approximate the solution(s) if necessary.
44.
Use graphing technology to solve the equation \(150x^2-20x+50=100x+40\text{.}\) Approximate the solution(s) if necessary.
45.
Use graphing technology to solve the inequality \(2x^2+5x-3\gt -5\text{.}\) State the solution set using interval notation, and approximate if necessary.
46.
Use graphing technology to solve the inequality \(-x^2+4x-7 \gt -12\text{.}\) State the solution set using interval notation, and approximate if necessary.
47.
Use graphing technology to solve the inequality \(10x^2-11x+7\leq7\text{.}\) State the solution set using interval notation, and approximate if necessary.
48.
Use graphing technology to solve the inequality \(-10x^2-15x+4\leq 9\text{.}\) State the solution set using interval notation, and approximate if necessary.
49.
Use graphing technology to solve the inequality \(-x^2-6x+1>x+5\text{.}\) State the solution set using interval notation, and approximate if necessary.
50.
Use graphing technology to solve the inequality \(3x^2+5x-4>-2x+1\text{.}\) State the solution set using interval notation, and approximate if necessary.
51.
Use graphing technology to solve the inequality \(-10x+4\leq 20x^2-34x+6\text{.}\) State the solution set using interval notation, and approximate if necessary.
52.
Use graphing technology to solve the inequality \(-15x^2-6\leq 10x-4\text{.}\) State the solution set using interval notation, and approximate if necessary.
53.
Use graphing technology to solve the inequality \(\frac{1}{2}x^2+\frac{3}{2}x \geq \frac{1}{2}x-\frac{3}{2}\text{.}\) State the solution set using interval notation, and approximate if necessary.
54.
Use graphing technology to solve the inequality \(\frac{3}{4}x\geq \frac{1}{4}x^2-3x\text{.}\) State the solution set using interval notation, and approximate if necessary.
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