Foundations Reference
Function Behavior
Function behavior describes what a graph does: whether it rises or falls, where it is positive or negative, what symmetry it has, and where it reaches high or low points.
Fact Table
| Behavior | Meaning | How to State It |
|---|---|---|
| Increasing | As $x$ moves right, $y$ goes up. | Use x-intervals. |
| Decreasing | As $x$ moves right, $y$ goes down. | Use x-intervals. |
| Constant | As $x$ moves right, $y$ stays the same. | Use x-intervals. |
| Positive | The graph is above the x-axis. | $f(x)>0$ |
| Negative | The graph is below the x-axis. | $f(x)<0$ |
| Zero | The graph touches or crosses the x-axis. | $f(x)=0$ |
| Even | Symmetric across the y-axis. | $f(-x)=f(x)$ |
| Odd | Symmetric through the origin. | $f(-x)=-f(x)$ |
Content Formulas
Even Test
$$f(-x)=f(x)$$
Odd Test
$$f(-x)=-f(x)$$
Positive and Negative
$$f(x)>0\quad\text{above the x-axis}$$ $$f(x)<0\quad\text{below the x-axis}$$
Zeros
$$f(x)=0$$
Increasing and decreasing intervals are always stated using x-values. We describe where the graph rises or falls, not the y-values it passes through.
Classic Examples
Even Function
Decide whether $f(x)=x^4-3x^2+7$ is even, odd, or neither.
Behavior MoveReplace $x$ with $-x$ and simplify.
$$f(-x)=(-x)^4-3(-x)^2+7$$
$$=x^4-3x^2+7$$
$$=f(x)$$
$$\text{even}$$
Odd Function
Decide whether $g(x)=x^3-5x$ is even, odd, or neither.
Behavior MoveAfter simplifying $g(-x)$, compare it with both $g(x)$ and $-g(x)$.
$$g(-x)=(-x)^3-5(-x)$$
$$=-x^3+5x$$
$$=-(x^3-5x)$$
$$=-g(x)$$
$$\text{odd}$$
Positive and Negative Intervals
Suppose a graph crosses the x-axis at $x=-2$ and $x=3$, is above the x-axis between those zeros, and below the x-axis outside them. State where $f(x)>0$ and $f(x)<0$.
Behavior MovePositive and negative intervals are x-intervals separated by zeros.
$$f(x)>0\text{ on }(-2,3)$$
$$f(x)<0\text{ on }(-\infty,-2)\cup(3,\infty)$$
Increasing and Decreasing
A graph rises until $x=1$, then falls after $x=1$. State the increasing and decreasing intervals.
Behavior MoveUse x-values. The turning point separates the intervals.
$$\text{increasing on }(-\infty,1)$$
$$\text{decreasing on }(1,\infty)$$
$$\text{local maximum at }x=1$$
Symmetry Summary Table
| Symmetry | Point Test | Equation Test |
|---|---|---|
| Y-axis symmetry | If $(x,y)$ is on the graph, then $(-x,y)$ is also on the graph. | Replace $x$ with $-x$. The equation stays equivalent. |
| X-axis symmetry | If $(x,y)$ is on the graph, then $(x,-y)$ is also on the graph. | Replace $y$ with $-y$. The equation stays equivalent. |
| Origin symmetry | If $(x,y)$ is on the graph, then $(-x,-y)$ is also on the graph. | Replace $x$ with $-x$ and $y$ with $-y$. The equation stays equivalent. |
| Even function | Y-axis symmetry for a function. | $f(-x)=f(x)$ |
| Odd function | Origin symmetry for a function. | $f(-x)=-f(x)$ |
Symmetry Formulas
Y-Axis Symmetry
$$(x,y)\Rightarrow(-x,y)$$
X-Axis Symmetry
$$(x,y)\Rightarrow(x,-y)$$
Origin Symmetry
$$(x,y)\Rightarrow(-x,-y)$$
Function Tests
$$\text{even: }f(-x)=f(x)$$ $$\text{odd: }f(-x)=-f(x)$$
X-axis symmetry usually means the relation is not a function of $x$, because most vertical lines would hit the graph twice.
Symmetry Classic Examples
Y-Axis Symmetry
Test $y=x^2$ for y-axis symmetry.
Symmetry MoveReplace $x$ with $-x$ and check whether the equation stays the same.
$$y=(-x)^2$$
$$=x^2$$
$$\text{y-axis symmetry}$$
X-Axis Symmetry
Test $x=y^2$ for x-axis symmetry. Is it a function of $x$?
Symmetry MoveReplace $y$ with $-y$. Then check the vertical line test.
$$x=(-y)^2$$
$$=y^2$$
$$\text{x-axis symmetry}$$
$$\text{not a function of }x$$
Origin Symmetry
Test $y=x^3$ for origin symmetry.
Symmetry MoveReplace both variables: $x$ with $-x$ and $y$ with $-y$.
$$-y=(-x)^3$$
$$-y=-x^3$$
$$y=x^3$$
$$\text{origin symmetry}$$
Several Symmetries
Test $x^2+y^2=25$ for x-axis, y-axis, and origin symmetry.
Symmetry MoveBecause both variables are squared, sign changes disappear.
$$(-x)^2+y^2=25$$
$$x^2+y^2=25$$
$$x^2+(-y)^2=25$$
$$x^2+y^2=25$$
$$(-x)^2+(-y)^2=25$$
$$x^2+y^2=25$$
$$\text{x-axis, y-axis, and origin symmetry}$$
Reading Graphs Checklist
| Feature | Question to Ask |
|---|---|
| Domain | What x-values are included? |
| Range | What y-values are included? |
| Zeros | Where does the graph meet the x-axis? |
| Positive/negative | Where is the graph above or below the x-axis? |
| Increasing/decreasing | Where does the graph rise or fall as we move left to right? |
| Extrema | Where are the local or absolute high and low points? |
| End behavior | What happens as $x\to\infty$ and $x\to-\infty$? |
| Symmetry | Does the graph have x-axis, y-axis, origin symmetry, or none of these? |