Please enable scripting (or JavaScript) in your web browser, and then reload this page.
In the last lesson, you learned how the graphs of $y=f(x)+k$ and $y=f(x-h)$ compare to the graph of $y=f(x)$. We’ll now look at the graphs of $y=af(x)$ and $$y=f(x/d)$$, to see how they compare to the graph of $y=f(x)$.
What are the values of $3f(x)$ at these values of $x$? Your answers will be plotted on the grid to the left.
Click to see the graphs of $y_1=f(x)$ and $y_2=3f(x)$.
Now the equation $y=af(x)$ is graphed on the grid to the left, with a slider for $a$.
In general:
If $a$ is positive, the graph of $y=af(x)$ has the same $x$-intercepts as the graph of $y_1=f(x)$ (it crosses the $x$-axis at the same places). At any other point, it is stretched $a$ times as far from the $x$-axis as the graph of $y_1=f(x)$.
Note that if $a<1$ (say, if $$a=1/2$$), the graph of $y=af(x)$ is compressed closer to the $x$-axis than the graph of $y_1=f(x)$. In this situation, we’ll still say that the graph is “stretched $a$ times as far from the $x$-axis as the graph of $y_1=f(x)$.”
Click to see the graphs of $y_1=f(x)$ and $y_2=-f(x)$.
Now the equation $y=af(x)$ is again graphed on the grid to the left, with a slider for $a$. This time, $a$ is allowed to be zero or negative.
You can think of multiplying $f(x)$ by a negative number $a$ as happening in two steps. First, multiply by the positive number $|a|$, which stretches the graph vertically. Then, multiply by $-1$, which flips the graph around the $x$-axis. This means:
If $a$ is negative, the graph of $y=af(x)$ has the same $x$-intercepts as the graph of $y_1=f(x)$ (it crosses the $x$-axis at the same places). At any other point, it is flipped across the $x$-axis and stretched $|a|$ times as far from the $x$-axis as the graph of $y_1=f(x)$.
The left-hand table below shows you the values of $g(x)$ for a certain function $\cl"red"g$, at a few values of $x$. We can find the value of $$g(x/2)$$ when $x=-4$ by first noticing that $${-4}/2=-2$$, and then using the left-hand table to find that $g(-2)=-3$. So, when $x=-4$, $$g(x/2)=-3$$.
Using the left-hand table in this way, what is the value of $$g(x/2)$$ at each value of $x$ in the right-hand table below?
Click to see the graphs of $y_1=g(x)$ and $$y_2=g(x/2)$$.
Now the equation $$y=g(x/d)$$ is graphed on the grid to the left, with a slider for $d$.
If $d$ is positive, the graph of $$y=f(x/d)$$ is stretched $d$ times as far from the $y$-axis as the graph of $y_1=f(x)$. The two graphs have the same $y$-intercepts (they meet the $y$-axis at the same places).
As before, if $d<1$, we still say that the graph of $$y=f(x/d)$$ is “stretched $d$ times as far from the $y$-axis” as the graph of $y_1=f(x)$, even though that means it is actually closer to the $y$-axis.
What about when $d$ is negative? We’ll start by looking at what happens when $d=-1$. In this case, we’re looking at the behavior of $$g(x/{-1})$$, which can be more simply written as $g(-x)$.
The left-hand table below again shows you the values of $g(x)$, at a few values of $x$. Using this table as in scale-by-2-qn, what is the value of $g(-x)$ at each value of $x$ in the right-hand table below?
Click to see the graphs of $y_1=g(x)$ and $y_2=g(-x)$.
Now the functions $y_1=g(x)$ and $$y=g(x/d)$$ are both graphed, with a slider for $d$. Notice that $d$ is always negative.
As in the previous section, you can think of the operation of dividing $x$ by the negative number $d$ as taking place in two steps. First you divide by the positive number $|d|$ (which stretches the graph horizontally). Then you multiply by $-1$ (which flips the graph across the $y$-axis). So:
If $d$ is negative, the graph of $$y=f(x / d)$$ is stretched $|d|$ times as far from the $y$-axis as the graph of $y_1=f(x)$, and flipped across the $y$-axis. The two graphs have the same $y$-intercepts (they meet the $y$-axis at the same places).
Now, we’ll look at the function $h(x)=x^2-2$. The graph of $y_1=h(x)$ is shown on the grid to the left, along with the graph of $y=ah(x)$, with a slider for $a$. As we saw in the first section above, a graph that is like the graph of $y_1=h(x)$, but stretched 3 times as far vertically, would have the equation $y=3h(x)$, or $y=3(x^2-2)=3x^2-6$.
Click to see the graph of $y_1=h(x)$ along with the graph of $$y=h(x/d)$$, with a slider for $d$. As we saw above in the second section of this lesson, a graph that is like the graph of $y_1=h(x)$, but stretched 2 times as far horizontally, would have the equation $$y=h(x/2)$$, or:
What would the equation be for a graph that was like the graph of $y_1=h(x)$, but stretched 3 times as far horizontally?
We saw above that the graph of $y=f(-x)$ can always be obtained by flipping the graph of $y=f(x)$ around the $y$-axis. Now, let’s see what happens if $f$ is a function whose graph doesn’t change when you flip it around the $y$-axis: that is, it has the $y$-axis as an axis of symmetry. In that case, $y=f(x)$ and $y=f(-x)$ have the same graph, so $f(x)=f(-x)$ for any $x$.
The function $f(x)=x^2$ is graphed on the grid to the left. Notice that the $y$-axis is an axis of symmetry of this graph. For each $x$ in the table below, compute $f(x)$ and $f(-x)$, and say whether they are equal.
In fact,
no matter what $x$ is, as we expected from looking at the graph of $\cl"red"f$. The same sort of calculation shows that $x^n=(-x)^n$ whenever $n$ is an even number. As a result, we call any function $f$ with $f(x)=f(-x)$ for all $x$ an even function.
Click to see a graph of $g(x)={|x|}-1$. This function is also even, because
no matter what $x$ is.
In fact:
A function $f$ will be even precisely when the graph of $f(x)$ has the $y$-axis as an axis of symmetry.
Remember that flipping the graph of $y=f(x)$ around the $x$-axis changes it into the graph of $y=-f(x)$, while flipping it around the $y$-axis changes it into the graph of $y=f(-x)$. Let’s see what happens when flipping a graph around the $x$-axis has the same effect as flipping it around the $y$-axis. That is, what if $f$ is a function where the graphs of $y=-f(x)$ and $y=f(-x)$ are the same? In that case, $-f(x)=f(-x)$ for any $x$, so $f(x)$ and $f(-x)$ are always opposites.
The function $f(x)=2x$ is graphed on the grid to the left, along with a flipped version in blue. Notice that you could flip the red graph around either the $x$-axis or the $y$-axis to get the blue graph.
For each $x$ in the table below, compute $f(x)$ and $f(-x)$, and say whether they are opposites.
no matter what $x$ is, as we expected from looking at the graphs.
Click to see a graph of $g(x)=x^3$, along with its flipped version in blue. This function also satisfies $-g(x)=g(-x)$, because
no matter what $x$ is. The same sort of calculation shows that $-x^n=(-x)^n$ whenever $n$ is an odd number. As a result, we call any function $f$ with $-f(x)=f(-x)$ for all $x$ an odd function.
In the last question, we saw that a function is odd if flipping it across either the $x$-axis or the $y$-axis has the same effect. There’s also another way to see whether a graph is the graph of an odd function, which is usually easier to see if the flipped version of the graph isn’t also shown.
Rotating by 180° around a point means rotating halfway (that is, turning upside-down) while holding that point fixed. For example, the blue graph on the left is the result of rotating the red graph by 180° around the point $(0,0)$.
If you can rotate a graph by 180° around some point and get back the same graph, we say that point is a point of symmetry of the function.
A function $f$ will be odd precisely when the graph of $f(x)$ has $(0,0)$ as a point of symmetry.
Some functions are neither even nor odd. For example, the function $f(x)=x^2+x$ is neither even nor odd. You can see this by looking at the graph of $f(x)$: the $y$-axis is not an axis of symmetry, and the origin is not a point of symmetry.
For each function in the table below, determine whether it is even, odd, or neither by using its graph.
You can also determine that a function is neither even nor odd by using its formula to find a number $x$ where $f(x)$ and $f(-x)$ are neither equal nor opposite. For example, to see that the function $f(x)=x^2+x$ is neither even nor odd, notice that $f(2)=2^2+2=4+2=6$, and $f(-2)=(-2)^2-2=4-2=2$. So $f(2)$ and $f(-2)$ are neither equal nor opposite, which means that $f$ is neither even nor odd.
Each function in the table below is neither even nor odd. Demonstrate this by giving some value of $x$ for which $f(x)$ and $f(-x)$ are neither equal nor opposite. Use scratch paper.