Section 5 Function Transformations and Rational Functions
In this section, we will examine how to build new functions from existing functions and how the graphs of the new functions relate to the original ones.
Task 5.1.
Given a function \(f(x)=x^2\) and two real numbers \(h\) and \(k\text{.}\)- Write down the function formula for a new function \(g(x) = f(x-h)+k\text{.}\)
- Go to the link https://www.desmos.com/calculator/z82wdytmxl. Begin by setting \(h = 0\) and \(k=0\text{.}\)
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The role of \(h\text{:}\)
- What happens as \(h\) increases from \(0\text{?}\)
- What happens as \(h\) decreases from \(0\text{?}\)
- Describes in words how different values of \(h\) affect the graph of the function \(g(x)\text{.}\)
- Without graphing, predict how the graph of \(q(x)=(x-4)^5\) will be different from the graph of \(p(x)=x^5\text{.}\)
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The role of \(k\text{:}\) (Remember to reset \(h=0\) and \(k=0\))
- What happens as \(k\) increases from \(0\text{?}\)
- What happens as \(k\) decreases from \(0\text{?}\)
- Describes in words how different values of \(k\) affect the graph of the function \(g(x)\text{.}\)
- Without graphing, predict how the graph of \(q(x)=x^5-4\) will be different from the graph of \(p(x)=x^5\text{.}\)
The role that \(h\) and \(k\) play can be summarized as horizontal and vertical shifts. Now we look at a different set of transformations.
Task 5.2.
Given a function \(f(x)=e^x\text{.}\)- Write down the function formula for two new functions \(g(x) = f(-x)\) and \(h(x) = -f(x)\text{.}\)
- Go to the link https://www.desmos.com/calculator/r5bvx2qhjr.Turn on the graphs of \(f(x)\) and \(g(x)\) in Desmos. How would you describe the procedure that obtains the graph of \(g(x)\) from the graph of \(f(x)\text{?}\)
- Turn on the graphs of \(f(x)\) and \(h(x)\) in Desmos. How would you describe the procedure that obtains the graph of \(h(x)\) from the graph of \(f(x)\text{?}\)
Shifts and reflections are called rigid transformations because the shape of the graph does not change. We now move to non-rigid transformations. But before that, we introduce a new function called the step function. The step function \(f(x) = \lfloor x \rfloor\) is defined to be the greatest integer less than or equal to \(x\text{.}\) The step function is also called the floor function.
Task 5.3.
Find \(\lfloor 1.5 \rfloor\text{,}\) \(\lfloor -1.5 \rfloor\) and \(\lfloor -10 \rfloor\text{.}\)Task 5.4.
Given a function \(f(x)=\lfloor x \rfloor\) and two real numbers \(a\) and \(b\text{.}\)- Write down the function formula for a new function \(g(x) = af(bx)\text{.}\)
- Go to the link https://www.desmos.com/calculator/qwuu0dswwf. Begin by setting \(a = 1\) and \(b = 1\text{.}\)
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The role of \(a\text{:}\)
- What happens as \(a\) increases from \(1\text{?}\)
- What happens as \(a\) decreases from \(1\) to \(0\text{?}\)
- What happen when \(a = -1\text{?}\)
- What happen as \(a\) increases from \(-1\) to \(0\text{?}\)
- What happen as \(a\) decreases from \(-1\text{?}\)
- Without graphing, predict how the graph of \(q(x)=-4x^5\) will be different from the graph of \(p(x)=x^5\text{.}\)
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The role of \(b\text{:}\) (Remember to reset \(a=1\) and \(b=1\))
- What happens as \(b\) increases from \(1\text{?}\)
- What happens as \(b\) decreases from \(1\) to \(0\text{?}\)
- What happen when \(b = -1\text{?}\)
- What happen as \(b\) increases from \(-1\) to \(0\text{?}\)
- What happen as \(b\) decreases from \(-1\text{?}\)
- Without graphing, predict how the graph of \(q(x)=(-4x)^5\) will be different from the graph of \(p(x)=x^5\text{.}\)
Task 5.5.
Given any function \(f(x)\text{,}\) describe how the graph of each function below will differ from the graph of \(f(x)\text{.}\) Try not to use any specific function equations to answer this question but if you are stuck, take \(f(x)=x^2\) and experiment it using Desmos.- \(\displaystyle 3f(x-2)\)
- \(\displaystyle f(3(x-2))\)
- \(\displaystyle -3f(x+2)-5\)
- \(\displaystyle 3f(-3(x-2))-7\)
- \(\displaystyle 5f(3(x+2))+7\)
A rational function is a quotient of polynomial functions that can be written as
where \(P(x), Q(x)\) are polynomials and \(Q(x)\) is not the zero polynomial. The domain of a rational function of \(x\) includes all real numbers except for a few \(x\)-values that make the denominator zero.
Task 5.6.
- Find the domain and the range of \(f(x) = \frac{1}{x}\text{.}\)
Fill in the table
Create a similar table for \(x\) approaching to \(0\) from the negative side. Where do you think \(f(x)\) approaches to when \(x \to 0\text{?}\)Table 5.7. Values of \(f(x)\) when \(x \to 0\) \(x\) \(1\) \(0.1\) \(0.01\) \(0.001\) \(f(x)\) \(\) \(\) \(\) \(\) Fill in the table
Where do you think \(f(x)\) approaches to when \(x \to +\infty\text{?}\) What about \(-\infty\text{?}\)Table 5.8. Values of \(f(x)\) when \(x \to +\infty\) \(x\) \(1\) \(10\) \(100\) \(1000\) \(f(x)\) \(\) \(\) \(\) \(\)
Definition 5.9.
We say that the line \(x=a\) is a vertical asymptote of the graph of \(f\) when \(f(x) \to \infty\) or \(f(x) \to -\infty\) as \(x \to a\text{,}\) either from the right or from the left. We say that the line \(y=b\) is a horizontal asymptote of the graph of \(f\) when \(f(x) \to b\) as \(x \to +\infty\) or \(x \to -\infty\text{.}\)Task 5.10.
Let \(f(x) = \frac{2x^2}{x^2-1}\text{.}\)- Find the domain and the range of \(f(x)\text{.}\)
- Find all the vertical and horizontal asymptotes of \(f(x)\text{.}\)
- Find the intervals that \(f(x)\) is increasing and the intervals that \(f(x)\) is decreasing.
- What is the average rate of change for \(3 \lt x \lt 6\text{?}\)
- Solve the inequality \(f(x) \gt 3\text{.}\)
Task 5.11.
Let \(f(x) = \frac{x^2+x-2}{x^2-x-6}\text{.}\)- Find the domain and the range of \(f(x)\text{.}\)
- Find all the vertical and horizontal asymptotes of \(f(x)\text{.}\)
- Find the intervals that \(f(x)\) is increasing and the intervals that \(f(x)\) is decreasing.
- What is the average rate of change for \(4 \lt x \lt 8\text{?}\)
- Solve the inequality \(f(x) \gt 6\text{.}\)