Key Features of
Functions
Learn to identify and sketch intercepts, extrema, intervals of increase and decrease, end behavior, and more from graphs and descriptions.
Algebra 2
What Are Key Features?
Every function's graph tells a story. Key features are the specific landmarks that help you read that story: where the graph crosses the axes, where it rises or falls, and where it reaches its highest or lowest points.
Intercepts
Where the graph crosses the \(x\)-axis and \(y\)-axis
Extrema
Relative (local) maximums and minimums
Behavior
Increasing/decreasing intervals and end behavior
Intercepts: Where the Graph Crosses the Axes
x-intercepts
Points where \(y = 0\). Set \(f(x)=0\) and solve. Also called zeros or roots.
y-intercept
The point where \(x = 0\). Evaluate \(f(0)\). There is at most one for a function.
Intervals: Increasing, Decreasing, Positive, Negative
Relative Maximums and Minimums
End Behavior: What Happens at the Extremes
Symmetry and Periodicity
Even Symmetry
Graph is symmetric about the \(y\)-axis. Algebraically: \(f(-x) = f(x)\). Example: \(f(x) = x^2\).
Odd Symmetry
Graph is symmetric about the origin. Algebraically: \(f(-x) = -f(x)\). Example: \(f(x) = x^3\).
The Complete Feature Checklist
1
Intercepts: find the \(x\)-intercepts (\(f(x)=0\)) and \(y\)-intercept (\(f(0)\)).
2
Increasing/Decreasing: identify intervals using \(x\)-values.
3
Positive/Negative: where is \(f(x) > 0\) vs. \(f(x) < 0\)?
4
Relative Extrema: locate local maximums and minimums.
5
End Behavior: describe \(f(x)\) as \(x \to \pm\infty\).
6
Symmetry/Periodicity: even, odd, or periodic pattern?
Absolute vs. Relative Extrema
| Absolute (Global) | Relative (Local) | |
|---|---|---|
| Definition | The single highest or lowest value over the entire domain | Highest or lowest in a nearby neighborhood |
| How many? | At most one max, one min | Can have several |
| Example | Vertex of a parabola opening up is an absolute min | Turning points of a cubic polynomial |
| On open domains? | May not exist | Can still exist |
How to Analyze a Graph Step by Step
1
Scan the axes. Mark every point where the graph touches or crosses the \(x\)-axis and the \(y\)-axis.
2
Trace left to right. Note where the graph goes up (increasing) and where it goes down (decreasing).
3
Mark the peaks and valleys. Label relative maximums and minimums with their coordinates.
4
Shade positive/negative. Note where the curve is above or below the \(x\)-axis.
5
Describe the ends. State what happens as \(x \to -\infty\) and \(x \to +\infty\).
Example 1: \(f(x) = -x^2 + 4x - 3\)
Example 2: Cubic Function
Example 3: Sketch from a Description
Common Mistakes to Avoid
Using y-values for intervals
Intervals where a function is increasing or decreasing are stated in terms of x-values, not y-values. Write \((1,3)\), not \((f(1), f(3))\).
Confusing positive with increasing
A function can be positive and decreasing at the same time (e.g., just past a maximum above the x-axis).
Forgetting endpoint behavior
End behavior is about what happens at the far left and far right. Do not confuse it with what happens near the y-axis.
Summary of Key Features
| Feature | How to Find | How to State |
|---|---|---|
| x-intercepts | Solve \(f(x) = 0\) | Ordered pairs \((a, 0)\) |
| y-intercept | Evaluate \(f(0)\) | Ordered pair \((0, b)\) |
| Increasing | Where the graph rises L \(\to\) R | Interval of x-values |
| Decreasing | Where the graph falls L \(\to\) R | Interval of x-values |
| Rel. Max/Min | Direction changes | Coordinate point \((x,y)\) |
| End Behavior | Leading term analysis | \(x \to \pm\infty\), \(f(x) \to ?\) |
Question 1 of 5
A function \(f\) has a relative maximum at the point \((3, 7)\). Which statement must be true?
A relative maximum means \(f(3) \geq f(x)\) for all \(x\) near 3. It does not have to be the absolute maximum, and the function changes from increasing to decreasing at that point, so it is not increasing there.
Question 2 of 5
On which interval is \(f(x) = x^2 - 6x + 5\) decreasing?
The vertex of \(f(x) = x^2 - 6x + 5\) is at \(x = \frac{-(-6)}{2(1)} = 3\). Since the parabola opens upward, the function decreases to the left of the vertex on \((-\infty, 3)\) and increases on \((3, +\infty)\).
Question 3 of 5
If \(f(-x) = -f(x)\) for all \(x\) in the domain, the function has:
The condition \(f(-x) = -f(x)\) is the definition of an odd function. Its graph has \(180^\circ\) rotational symmetry about the origin. Even symmetry would require \(f(-x) = f(x)\).
Question 4 of 5
A polynomial function has a positive leading coefficient and odd degree. What is its end behavior?
For odd degree with a positive leading coefficient, the graph falls to the left (\(f(x) \to -\infty\)) and rises to the right (\(f(x) \to +\infty\)). Think of \(f(x) = x^3\).
Question 5 of 5
The graph of a function crosses the \(x\)-axis at \((-1,0)\) and \((5,0)\), has a \(y\)-intercept at \((0,10)\), and has a relative maximum at \((2,18)\). On which interval is the function positive?
The function is positive where \(f(x) > 0\), which means the graph is above the \(x\)-axis. Since the \(x\)-intercepts are at \(x = -1\) and \(x = 5\), and the function reaches \(18\) between them, the function is positive on \((-1, 5)\). Remember: state in terms of \(x\)-values!