Analyzing Functions Using Graphs: College Algebra Review

Functions

Analyzing Functions Using Graphs

Understanding how to interpret and analyze the graph of a function is a fundamental skill in College Algebra. Graphs provide visual information about the behavior, intercepts, and range of a function.

• Function: A relation in which each input (x-value) has exactly one output (y-value).

• Graph of a Function: The set of all points (x, f(x)) in the coordinate plane.

Key Concepts and Steps

• Intercepts:

  • x-intercept(s): The point(s) where the graph crosses the x-axis. At these points, f(x) = 0.

  • y-intercept: The point where the graph crosses the y-axis. At this point, x = 0.

• Increasing and Decreasing Intervals:

  • The function is increasing on intervals where the graph rises as x increases.

  • The function is decreasing on intervals where the graph falls as x increases.

• Constant Intervals: The function is constant on intervals where the graph is a horizontal line (f(x) does not change as x changes).

• Range: The set of all possible output values (y-values) of the function.

Example: Analyzing a Function from Its Graph

1. Find the x-intercepts: Identify all points where the graph crosses the x-axis (y = 0).

2. Find the y-intercept: Identify the point where the graph crosses the y-axis (x = 0).

3. Determine intervals of increase, decrease, and constancy:

   • Look for sections where the graph moves upward (increasing), downward (decreasing), or remains flat (constant).

4. Determine the range: Observe the lowest and highest y-values the graph attains.

Sample Table: Intervals of a Function

Interval	Behavior
(a, b)	Increasing
(b, c)	Decreasing
(c, d)	Constant

Additional info: The specific intervals (a, b), (b, c), (c, d) should be determined by analyzing the graph provided in the question.

Formulas and Notation

• Function notation: $f(x)$

• x-intercept: $f(x) = 0$

• y-intercept: $f(0)$

• Increasing interval: $f(x_1) < f(x_2)$ for $x_1 < x_2$ in the interval

• Decreasing interval: $f(x_1) > f(x_2)$ for $x_1 < x_2$ in the interval

• Constant interval: $f(x_1) = f(x_2)$ for all $x_1, x_2$ in the interval

Example Application

• Given a graph, suppose the function crosses the x-axis at $x = -2, 1, 3$. These are the x-intercepts.

• If the graph crosses the y-axis at $y = 4$, then the y-intercept is $f(0) = 4$.

• If the graph rises from $x = -$ to $x = -2$, the function is increasing on $(-5, -2)$.

• If the graph falls from $x = -2$ to $x = 1$, the function is decreasing on $(-2, 1)$.

• If the graph is flat from $x = 1$ to $x = 2$, the function is constant on $(1, 2)$.

• The range is the set of all y-values the graph attains, e.g., $[-3, 5]$.