Functions 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.

Solved Examples

Jump directly to a worked example by behavior or symmetry type.

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. $f(-x)=-f(x)$
Polynomial end behavior Look at the degree and the sign of the leading term. Even degree: same direction. Odd degree: opposite directions.

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
$$f(x)=0$$
Polynomial End Behavior
$$ax^n,\ n\text{ even} \Rightarrow \text{same end direction}$$ $$ax^n,\ n\text{ odd} \Rightarrow \text{opposite end directions}$$ $$a\gt0:\ \text{right end up}\qquad a\lt0:\ \text{right end down}$$
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.

Polynomial End Behavior Quick Table

For polynomials, the leading term controls the ends. Higher degree can add more turns in the middle, but the far-left and far-right directions still come from the degree parity and the sign of the leading coefficient.

Leading Term Type Left End Right End
$+x^2,\ +x^4,\ +x^6$ Up Up
$-x^2,\ -x^4,\ -x^6$ Down Down
$+x^3,\ +x^5,\ +x^7$ Down Up
$-x^3,\ -x^5,\ -x^7$ Up Down
Shortcut: even degree means both ends match; odd degree means the ends oppose each other. The sign of the leading coefficient tells whether the right end finishes up or down.

Graph Cues

These sketches are not full graphing exercises. They are fast visual cues for the most common behavior questions students are asked to recognize and compare.

Decreasing, Increasing

Read the intervals from left to right using x-values. The turning point separates the increasing and decreasing parts.

Positive, Negative, Zero

Positive means above the x-axis. Negative means below it. Zeros are where the graph touches or crosses the axis.

Even Symmetry

An even function mirrors across the y-axis, so matching x-values with opposite signs give the same output.

Odd Symmetry

An odd function has origin symmetry. If one point is on the graph, the opposite point through the origin is also on the graph.

Local Minimum

Extrema are special points where the graph switches direction. A local minimum is lower than nearby points.

End Behavior

End behavior describes what the graph does far to the left and far to the right, not just near the center of the picture.

Classic Examples

Even Function

Decide whether $f(x)=x^4-3x^2+7$ is even, odd, or neither.

Solution Steps
  1. Replace $x$ with $-x$.
  2. Simplify the even powers.
  3. Compare the result with $f(x)$.
  4. State that the function is even.
$$f(-x)=(-x)^4-3(-x)^2+7$$ $$=x^4-3x^2+7$$ $$=f(x)$$ $$\text{even}$$

Odd Function

Decide whether $g(x)=x^3-5x$ is even, odd, or neither.

Solution Steps
  1. Replace $x$ with $-x$.
  2. Simplify the powers and products.
  3. Factor out the negative sign.
  4. Compare the result with $-g(x)$.
  5. State that the function is odd.
$$g(-x)=(-x)^3-5(-x)$$ $$=-x^3+5x$$ $$=-(x^3-5x)$$ $$=-g(x)$$ $$\text{odd}$$

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 $f(x)<0$.

Solution Steps
  1. Use the zeros to split the x-axis into intervals.
  2. Read where the graph is above and below the x-axis, then state both intervals.
$$f(x)>0\text{ on }(-2,3)$$ $$f(x)<0\text{ on }(-\infty,-2)\cup(3,\infty)$$

Increasing and Decreasing

A graph rises until $x=1$, then falls after $x=1$. State the increasing and decreasing intervals.

Solution Steps
  1. Read the interval where the graph rises.
  2. Read the interval where the graph falls.
  3. Identify the local maximum at the turning point.
$$\text{increasing on }(-\infty,1)$$ $$\text{decreasing on }(1,\infty)$$ $$\text{local maximum at }x=1$$

Polynomial End Behavior

Describe the end behavior of $f(x)=-2x^5+4x^2-1$.

Solution Steps
  1. Identify the leading term.
  2. Use its odd degree and negative coefficient to determine end directions.
  3. Evaluate the left-end behavior.
  4. Evaluate the right-end behavior.
  5. State the end behavior in words.
$$\text{leading term }=-2x^5$$ $$\text{odd degree with negative leading coefficient}$$ $$x\to-\infty\Rightarrow f(x)\to\infty$$ $$x\to\infty\Rightarrow f(x)\to-\infty$$ $$\text{left end up, right end down}$$

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.

Solution Steps
  1. Replace $x$ with $-x$.
  2. Simplify the equation.
  3. State the y-axis symmetry result.
$$y=(-x)^2$$ $$=x^2$$ $$\text{y-axis symmetry}$$

X-Axis Symmetry

Test $x=y^2$ for x-axis symmetry. Is it a function of $x$?

Solution Steps
  1. Replace $y$ with $-y$.
  2. Simplify the equation.
  3. State the x-axis symmetry result.
  4. Apply the vertical line test to classify the relation.
$$x=(-y)^2$$ $$=y^2$$ $$\text{x-axis symmetry}$$ $$\text{not a function of }x$$

Origin Symmetry

Test $y=x^3$ for origin symmetry.

Solution Steps
  1. Replace $x$ with $-x$ and $y$ with $-y$.
  2. Simplify the transformed equation.
  3. Return to the original equation.
  4. State the origin symmetry result.
$$-y=(-x)^3$$ $$-y=-x^3$$ $$y=x^3$$ $$\text{origin symmetry}$$

Several Symmetries

Test $x^2+y^2=25$ for x-axis, y-axis, and origin symmetry.

Solution Steps
  1. Replace $x$ with $-x$ and simplify.
  2. Confirm y-axis symmetry.
  3. Replace $y$ with $-y$ and simplify.
  4. Confirm x-axis symmetry.
  5. Replace both variables with their opposites.
  6. Confirm origin symmetry.
  7. State all three symmetry types.
$$(-x)^2+y^2=25$$ $$x^2+y^2=25$$ $$x^2+(-y)^2=25$$ $$x^2+y^2=25$$ $$(-x)^2+(-y)^2=25$$ $$x^2+y^2=25$$ $$\text{x-axis, y-axis, and origin symmetry}$$

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?