A.10(E)·Algebra II·Rational FunctionsSupporting

Rational Function Graphs

Determine where rational-function values satisfy an inequality.

Students solve rational inequalities from graphs and tables by locating zeros and vertical asymptotes, testing each interval, and describing where $f(x)>0$ or $f(x)<0$ .

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Read the inequality from the graph

$f(x)>0$ means the output is positive, so the graph lies above the x-axis. $f(x)<0$ means the graph is below the x-axis. The answer is a set of x-values.

Find boundary points

A rational function can change sign at a zero, where its numerator is 0, or at a vertical asymptote, where its denominator is 0. For $\frac{x-2}{x+1}$ , the boundary points are $x=2$ and $x=-1$ .

Test and exclude

Choose a test value in each interval to find the sign. For strict inequalities, exclude zeros. Always exclude vertical asymptote values because the function is undefined there.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 8 · 1 point

A rational graph has vertical asymptotes at x equals negative 2 and x equals 2. Its upper branches are above the x-axis for x less than negative 2 or greater than 2; the middle branch is below it. — Rational function fOriginal visual from the released STAAR test.

Worked example

For $f(x)=\frac{x-2}{x+1}$ , find where $f(x)>0$ .

Boundary points: the zero is $x=2$ and the asymptote is $x=-1$ .
Test $x=-2$ : $\frac{-4}{-1}=4>0$ .
Test $x=0$ : $\frac{-2}{1}=-2<0$ .
Test $x=3$ : $\frac{1}{4}>0$ .

$x<-1$ or $x>2$

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Quadratic Inequalities

UT IPSI · Kid2Kid / Texas Education Agency

Use graphs and intervals to decide where a function is above or below the x-axis.

Watch on YouTube

Texas public school video

Algebra 2 Identifying Parent Functions

UT IPSI · Kid2Kid / Texas Education Agency

Recognize parent-function graphs, including rational functions, by their key features.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. A graph of g has a vertical asymptote at x = 3. It is below the x-axis for x < 3 and above it for x > 3. Solve g(x) > 0.
Show answer and reasoning
$x>3$
The graph is above the x-axis only to the right of the asymptote. The value 3 is excluded because g is undefined there.
2. For $h(x)=\frac{1}{x}$ , solve $h(x)<0$ .
Show answer and reasoning
$x<0$
One divided by a negative number is negative. The function is undefined at 0 and positive for every positive x.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 8 · 1 point

The graph of the function $f$ is shown on the grid. Based on the graph, for what values of $x$ is $f(x) > 0$ ?

All real numbers greater than 0
All real numbers greater than 1
All real numbers less than −2 or greater than 2
All real numbers greater than −2 and less than 2

Show STAAR answer and explanation

All real numbers less than −2 or greater than 2

Positive function values lie above the x-axis. The left upper branch has $x<-2$ , and the right upper branch has $x>2$ .

All real numbers greater than 0: This includes values between 0 and 2, where the middle branch is below the x-axis, and omits the positive left branch.

All real numbers greater than 1: This includes $1<x<2$ , where the graph is negative, and omits the positive left branch.

All real numbers greater than −2 and less than 2: The middle branch is below the x-axis on this interval, so these values give $f(x)<0$ .

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Interpret $f(x)>0$ as the part of a graph above the x-axis.
Use zeros and vertical asymptotes to divide the x-axis into intervals.
Write solution sets with inequalities, excluding values where f is undefined or equal to zero.

Instructional sequence

1
Engage
5 min
Shade the parts of a rational graph that lie above the x-axis. Ask what x-values those pieces use.
2
Direct instruction
12 min
Model finding zeros and vertical asymptotes, then marking them as boundary points on a number line.
3
Sign chart practice
13 min
Test one value in each interval and record whether the output is positive or negative.
4
Graph and algebra
13 min
Match sign-chart results to the graph and write each answer with inequality notation.
5
Closure
7 min
Students explain why an asymptote value is never included, then complete the exit ticket.

Misconceptions to address

““f(x) > 0 means x > 0.””

Teacher move: The inequality describes the output. Look for where the graph is above the x-axis, which can include negative x-values.

““The vertical asymptote value belongs in the solution.””

Teacher move: The function is undefined there, so the value is excluded even if the graph is positive on both sides.

Materials

•Rational-function graph cards
•Sign-chart template
•Graphing calculator

Teaching tips

Mark zeros with open or closed points and asymptotes with dashed lines before shading.
Require one test value per interval instead of guessing signs from a sketch.
Have students state each answer in words: “The graph is above the x-axis when …”

Exit ticket · 3–5 min

For $f(x)=\frac{x+4}{x-1}$ , find where $f(x)<0$ .

Show expected answer

The zero is $x=-4$ and the asymptote is $x=1$ . Testing $x=0$ gives $\frac{4}{-1}=-4<0$ , so $-4<x<1$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Read the inequality from the graph

Find boundary points

Test and exclude

Study released STAAR visuals

Archived released STAAR Algebra II EOC · 2013 · Item 8 · 1 point

Worked example

Watch the lesson

Algebra 2 Quadratic Inequalities

Algebra 2 Identifying Parent Functions

Try it yourself

1. A graph of g has a vertical asymptote at x = 3. It is below the x-axis for x < 3 and above it for x > 3. Solve g(x) > 0.

2. For h(x)=1xh(x)=\frac{1}{x}h(x)=x1​, solve h(x)<0h(x)<0h(x)<0.

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 8 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Sign chart practice

Graph and algebra

Closure

Misconceptions to address

2. For $h(x)=\frac{1}{x}$ , solve $h(x)<0$ .