Learn on PengiCalifornia Reveal Math, Algebra 1Unit 2: Relations and Functions

2-6 Sketching Graphs and Comparing Functions

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to sketch graphs of linear and nonlinear functions using key features such as intercepts, relative maxima and minima, increasing and decreasing intervals, and end behavior. Students also practice interpreting these features in real-world contexts and comparing the properties of two functions. The lesson is part of Unit 2: Relations and Functions and builds graph-reading and function analysis skills essential for Algebra 1.

Section 1

Sketching Graphs Using Key Features

Property

A graph can be sketched by combining key features: intercepts, increasing/decreasing intervals, positive/negative intervals, and end behavior.

To sketch a graph:

Plot the $x$ - and $y$ -intercepts.
Identify where the function is increasing or decreasing.
Identify where the function is positive (above the $x$ -axis) or negative (below the $x$ -axis).
Apply the end behavior to determine what happens as $x \to -\infty$ and $x \to +\infty$ .

Section 2

Comparing Key Features of Two Functions

Property

To compare two functions $f(x)$ and $g(x)$ , build a side-by-side table of their key features:

\begin{array}{|l|c|c|} \hline \textbf{Feature} & f(x) & g(x) \\ \hline \text{y-intercept} & f(0) & g(0) \\ \text{x-intercept(s)} & f(x)=0 & g(x)=0 \\ \text{Relative max/min} & (x_1,\, f(x_1)) & (x_2,\, g(x_2)) \\ \text{Increasing on} & (a,\, b) & (c,\, d) \\ \text{Decreasing on} & (b,\, e) & (d,\, h) \\ \text{End behavior} & \text{describe} & \text{describe} \\ \hline \end{array}

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Sketching Graphs Using Key Features

Property

A graph can be sketched by combining key features: intercepts, increasing/decreasing intervals, positive/negative intervals, and end behavior.

To sketch a graph:

Plot the $x$ - and $y$ -intercepts.
Identify where the function is increasing or decreasing.
Identify where the function is positive (above the $x$ -axis) or negative (below the $x$ -axis).
Apply the end behavior to determine what happens as $x \to -\infty$ and $x \to +\infty$ .

Section 2

Comparing Key Features of Two Functions

Property

To compare two functions $f(x)$ and $g(x)$ , build a side-by-side table of their key features:

\begin{array}{|l|c|c|} \hline \textbf{Feature} & f(x) & g(x) \\ \hline \text{y-intercept} & f(0) & g(0) \\ \text{x-intercept(s)} & f(x)=0 & g(x)=0 \\ \text{Relative max/min} & (x_1,\, f(x_1)) & (x_2,\, g(x_2)) \\ \text{Increasing on} & (a,\, b) & (c,\, d) \\ \text{Decreasing on} & (b,\, e) & (d,\, h) \\ \text{End behavior} & \text{describe} & \text{describe} \\ \hline \end{array}

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Sketching Graphs Using Key Features

Property

A graph can be sketched by combining key features: intercepts, increasing/decreasing intervals, positive/negative intervals, and end behavior.

To sketch a graph:

Plot the $x$ - and $y$ -intercepts.
Identify where the function is increasing or decreasing.
Identify where the function is positive (above the $x$ -axis) or negative (below the $x$ -axis).
Apply the end behavior to determine what happens as $x \to -\infty$ and $x \to +\infty$ .

Section 2

Comparing Key Features of Two Functions

Property

To compare two functions $f(x)$ and $g(x)$ , build a side-by-side table of their key features:

\begin{array}{|l|c|c|} \hline \textbf{Feature} & f(x) & g(x) \\ \hline \text{y-intercept} & f(0) & g(0) \\ \text{x-intercept(s)} & f(x)=0 & g(x)=0 \\ \text{Relative max/min} & (x_1,\, f(x_1)) & (x_2,\, g(x_2)) \\ \text{Increasing on} & (a,\, b) & (c,\, d) \\ \text{Decreasing on} & (b,\, e) & (d,\, h) \\ \text{End behavior} & \text{describe} & \text{describe} \\ \hline \end{array}