Unit 2: Characteristics of Graphs

Unit 2 Resources & Lessons

VA SOL Standard: AFDA.AF.2. This unit focuses on analyzing graphs to understand their key features. We will use the Desmos Graphing Calculator to find these features visually.

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Daily Lesson Slides

Day 1: Domain and Range

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Day 2: Intercepts and Zeros

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Day 3: Increasing & Decreasing Intervals

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Day 4: End Behavior & Asymptotes

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Unit 2 Worksheets

Unit 2 Practice Worksheet Unit 2 Study Guide

Day 1: Domain and Range

Objective: To find the domain and range of a function by looking at its graph.

Notes

Domain: A list of all possible x-values (left to right) that the graph covers.

Range: A list of all possible y-values (bottom to top) that the graph covers.

x y Domain Range
Desmos Exploration: Visualizing Domain and Range
  1. In Desmos, graph the equation y = (x-2)^2 + 1.
  2. For Domain: The arrows show it goes forever left and right. The domain is "All Real Numbers," or $(-\infty, \infty)$.
  3. For Range: The lowest y-value it touches is 1. From there, it goes up forever. The range is "y is greater than or equal to 1," or $[1, \infty)$.

Day 2: Intercepts and Zeros

Objective: To find the x-intercepts, y-intercepts, and zeros of a function by clicking on its graph in Desmos.

Notes

Y-Intercept: The point where the graph crosses the vertical y-axis.

X-Intercepts: The points where the graph crosses the horizontal x-axis.

Zeros: This is just another name for the x-intercepts. Finding the "zeros" means finding where the graph's height is zero.

Desmos Exploration: Finding Intercepts
  1. In Desmos, graph the equation y = x^2 - 4.
  2. Click on the points where the graph crosses the axes.
  3. Desmos will show gray dots with the coordinates. The point on the y-axis, $(0, -4)$, is the y-intercept. The points on the x-axis, $(-2, 0)$ and $(2, 0)$, are the x-intercepts (or zeros).

Day 3: Increasing & Decreasing Intervals

Objective: To identify the parts of a graph that are increasing (going uphill) or decreasing (going downhill).

Notes

We always read a graph from left to right.

Increasing Interval: Any section of the graph that is going uphill.

Decreasing Interval: Any section of the graph that is going downhill.

We describe these sections using their x-values. The point where the graph changes direction is called a turning point or vertex.

Desmos Exploration: Finding Intervals
  1. Graph y = -x^2 + 2x + 3 in Desmos.
  2. Trace the graph from the left. It goes uphill until its peak. This is the increasing interval.
  3. After the peak, the graph goes downhill. This is the decreasing interval.
  4. Click on the peak (the vertex). Desmos shows its coordinates are $(1, 4)$. The x-value, $1$, is where the graph changes. It's increasing when $x < 1$ and decreasing when $x > 1$.

Day 4: End Behavior & Asymptotes

Objective: To describe where the arrows of a graph are pointing (end behavior) and identify invisible boundary lines (asymptotes).

Notes

End Behavior: Describes what the arrows at the very ends of the graph are doing. We ask: as $x$ goes to the far left (to $-\infty$), where does $y$ go? As $x$ goes to the far right (to $\infty$), where does $y$ go?

Asymptote: An invisible boundary line that a graph approaches but never touches. This often happens with exponential graphs.

Desmos Exploration: Seeing End Behavior
  1. Graph y = -x^2 + 5. The left arrow points down (to $-\infty$) and the right arrow points down (to $-\infty$).
  2. Now, graph y = 2^x. The right arrow points up (to $\infty$).
  3. The left side of the y=2^x graph gets very close to the x-axis ($y=0$) but never touches it. That horizontal line at $y=0$ is an asymptote.