Analyzing a Piecewise Function from Its Graph

Q2. Consider the following graph of $y = f(x)$:

Parts:

• (a) Find:

  • x-intercepts

  • y-intercept

• (b) Relative Extrema:

  • Maximums

  • Minimums

• (c) Intervals where the function is:

  • Increasing

  • Decreasing

  • Constant

• (d) Find:

  • $f(2)$

  • $f(0)$

Background

Topic: Graphical Analysis of Functions

This question tests your ability to interpret a function's graph, including identifying intercepts, extrema, intervals of increase/decrease/constant, and evaluating the function at specific points.

Key Terms:

• x-intercept: The point(s) where the graph crosses the x-axis ($y = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• Relative maximum/minimum: Highest/lowest points in a local region of the graph.

• Increasing interval: Where the graph moves upward as $x$ increases.

• Decreasing interval: Where the graph moves downward as $x$ increases.

• Constant interval: Where the graph is flat (horizontal).

Step-by-Step Guidance

1. To find the x-intercepts, look for points where the graph crosses the x-axis. These are points where $y = 0$.

2. To find the y-intercept, locate the point where the graph crosses the y-axis. This is where $x = 0$.

3. For relative extrema, identify the highest and lowest points in each segment of the graph. These are local maximums and minimums.

4. To determine intervals of increase, decrease, and constant, observe the direction of the graph as $x$ increases. If the graph goes up, it's increasing; if it goes down, it's decreasing; if it's flat, it's constant.

5. To evaluate $f(2)$ and $f(0)$, find the corresponding $y$ values for $x = 2$ and $x = 0$ on the graph.

Try solving on your own before revealing the answer!