Last updated October 14, 2026.

Secondary Math III · Module 4 · Rational Expressions & Functions

4.1 — Winner, Winner

Supplement for Mathematics Vision Project (MVP) curriculum · Secondary Math III, Module 4, Section 4.1
Independently created to support students using MVP Math. Not official publisher material and not affiliated with or endorsed by the Mathematics Vision Project.

Human review completed October 14, 2026.

Explore the graph of the reciprocal function, its two branches, and its vertical and horizontal asymptotes.

1. In class: The graph of the reciprocal function

The reciprocal function is $f(x)=\frac1x$.

The graph below has two separate branches. Positive inputs give positive outputs, so one branch lies in Quadrant I. Negative inputs give negative outputs, so the other lies in Quadrant III. At $x=0$, $f(0)$ is undefined because we cannot divide by zero, so there is no point on the graph with an input of 0. Use the table to locate points on each branch.

The graph of $y=\frac1x$

x	$-4$	$-2$	$-1$	$-\frac12$	$\frac12$	$1$	$2$	$4$
f(x)	$-\frac14$	$-\frac12$	$-1$	$-2$	$2$	$1$	$\frac12$	$\frac14$

$f(-x)=-f(x)$, so matching negative inputs have opposite outputs. The function is odd: rotating the first-quadrant branch half a turn around the origin gives the third-quadrant branch.

Main 12 graph: the two branches stay separate at x=0. The marked points are $(1,1),(-1,-1),(\frac12,2),(-\frac12,-2)$.

The dashed lines are the asymptotes $x=0$ and $y=0$. Equal x- and y-unit lengths.

Why does the graph get so close to the axes?

Look at the right-hand branch. As you move farther to the right, the graph gets closer and closer to the x-axis. For example, $\frac1{10}=0.1$, $\frac1{100}=0.01$, and $\frac1{1000}=0.001$. The outputs get closer to 0, but never equal 0. The left-hand branch also gets closer to the x-axis as you move farther to the left.

We call the line $y=0$, which is the x-axis, a horizontal asymptote. It tells us which horizontal line the graph gets closer to as we move far to the left or right.

Now look near the y-axis. Try positive inputs that get closer to 0:

x	$y=\frac1x$
0.1	10
0.01	100
0.001	1000

The closer these inputs get to 0, the higher the graph climbs. On the other side, $\frac1{-0.1}=-10$ and $\frac1{-0.01}=-100$: the graph drops farther and farther down as negative inputs get closer to 0.

The line $x=0$, which is the y-axis, is a vertical asymptote. As the graph gets closer to this line, it keeps climbing or dropping without stopping. There is no point on the graph at $x=0$, because we cannot divide by 0.

Feature	Answer and reason
Domain	$(-\infty,0)\cup(0,\infty)$; x=0 would divide by zero.
Range	$(-\infty,0)\cup(0,\infty)$; every nonzero y occurs at $x=\frac1y$.
Increase / decrease	Decreasing on $(-\infty,0)$ and separately on $(0,\infty)$; no interval of increase.
Intercepts	No x-intercept (output never zero), no y-intercept (x=0 excluded).
End behavior	Far to the right, outputs get closer to 0 from above. Far to the left, they get closer to 0 from below.
Maxima / minima	None: each branch keeps decreasing and outputs are unbounded above and below.

Teacher’s guide correction — Section 4.1, Question 13

Question 13 in the student packet (printed p. 4) asks students to describe the features of $f(x)=\frac1x$, including its intervals of increase or decrease. In the MVP Secondary Math III, Module 4: Rational Expressions & Functions teacher’s guide, Section 4.1 “Winner, Winner,” the closing paragraph of “Discuss (Whole Group)” (PDF p. 12) says:

“Be sure that students can use proper notation to describe the domain and range, as well as recognizing that the function is decreasing through the entire domain.”

Why this wording is incorrect: Saying a function is decreasing over its entire domain means that whenever we choose a larger input, we get a smaller output. That fails when we compare inputs on opposite sides of 0. For example, moving from $x=-1$ to $x=1$ makes the output go from −1 to 1: the input increases, but the output increases too. Both inputs are in the domain, so this is a counterexample to the teacher’s guide’s statement.

Use this wording instead: “The function is decreasing on $(-\infty,0)$ and decreasing on $(0,\infty)$.” In other words, each branch goes downhill as you read it from left to right. Describe the two intervals separately because the graph has a break at $x=0$.

2. Ready — Read quadratic transformations

In your MVP packetSection 4.1 · Ready 1–6 · printed p. 5

For a refresher, see Math II 2.1: quadratic transformations. Vertex form is $y=a(x-h)^2+k$: the vertex is (h,k), the sign of a determines opening direction, and |a| controls vertical scaling. Read the vertex, then use another visible point to find a. Try the six graphs before checking.

Check your work — Ready 1–6

Problem	Transformation from $y=x^2$	Vertex form
1	Down 6	$y=x^2-6$
2	Down 5	$y=x^2-5$
3	Left 3	$y=(x+3)^2$
4	Reflect across x-axis, right 2, down 1	$y=-(x-2)^2-1$
5	Vertical compression by $\frac12$, left 3, up 2	$y=\frac12(x+3)^2+2$
6	Reflect across x-axis, left 3, up 5	$y=-(x+3)^2+5$

In 5, the vertex is (−3,2) and the graph passes through (−5,4), so $4=a(-5+3)^2+2$ gives $a=\frac12$.

3. Set — Share a $360 gift

In your MVP packetSection 4.1 · Set 7–10 · printed p. 6

Use $C(n)=\frac{360}{n}$, where n includes everyone contributing, including Hannah. The output is dollars per person. Study the sharing model above, then try Set.

Check your work — Set 7–10

Problem	Answer
7	2 people: $180 each; 5: $72; 10: $36; 100: $3.60.
8	Numerator: total gift cost, $360. Denominator: number of contributors.
9	Sample table: (1,360), (2,180), (5,72), (10,36), (100,3.60). Plot not shown: label people horizontally and dollars per person vertically; use positive integer inputs and separate points.
10a	$n=\frac{360}{0.50}=720$ total contributors.
10b	$n=\frac{360}{0.10}=3600$ total contributors, including Hannah. If counting only her other friends, there are 3599.

4. Go — Match exponential graphs and asymptotes

In your MVP packetSection 4.1 · Go 11–18 · printed p. 7

For $y=ab^{x-h}+k$, with $a\ne0$, $b>0$ and $b\ne1$, the horizontal asymptote is $y=k$. Changing h moves the graph horizontally; a scales or reflects it. Review Math II 3.1: exponential functions. Use an asymptote and a convenient point to distinguish the packet’s graphs, then check.

Check your work — Go 11–18

Problem / rule	Graph letter	Horizontal asymptote
11	$c$	$y=0$
12	f	$y=-3$
13	e	$y=0$
14	b	$y=-3$
15	a	$y=3$
16	d	$y=0$

Checks: 11 has (0,1); 12 has (0,−2); 13 has (3,1); 14 has (0,−4); 15 has (0,4); 16 has (0,−1).

17: k sets the position of the horizontal asymptote $y=k$. The other parameters do not move that line. The base b determines which end approaches it.

18: for $b>1$, $b^{x-h}\to0$ as $x\to-\infty$; for $0<b<1$, it approaches zero as $x\to\infty$. Multiplying by fixed a still gives a term approaching zero; adding k makes the output approach k.

Additional learning resources

Resource	What it helps with and scope
Khan Academy — Intro to direct & inverse variation	The inverse-variation part connects reciprocal tables, scaling and a fixed product; the video also compares direct variation.

Explore asymptotes more fully

These optional videos focus on asymptotes. They cover more than this individual lesson, including general rational functions and ideas developed elsewhere in Module 4. Start with the first video for the basic idea; use the others to explore vertical asymptotes, holes, and end behavior in more depth.

Resource	What it helps with and scope
Khan Academy — Graphing rational functions 1	Start here for an explanation of vertical and horizontal asymptotes using a graph and nearby input values. Broader than this lesson: uses a shifted rational function rather than just 1/x and also finds intercepts.
Khan Academy — Graphs of rational functions: vertical asymptotes	More practice connecting a rational expression to its vertical asymptotes. Broader than this lesson: examples involve factored expressions beyond the basic reciprocal graph.
Khan Academy — Graphing rational functions according to asymptotes	Explains why an excluded input may produce a hole rather than a vertical asymptote, and identifies horizontal asymptotes. Broader than this lesson: combines factoring, cancellation, and several graph features; useful as a preview or later review.
Khan Academy — Analyzing horizontal asymptotes of rational functions	A deeper explanation of horizontal asymptotes through end behavior and leading terms. Broader than this lesson: compares multiple rational functions and degree cases rather than focusing on one graph.

Teacher/parent review of the source packet

Teacher’s guide, Section 4.1, Question 13: The “Discuss (Whole Group)” guidance (PDF p. 12) incorrectly describes the function as decreasing through its entire domain. See the quoted passage, counterexample, and corrected wording above.

Equal sharing provides a starting example of $y=1/x$, but the task introduces complications as it extends the story. In Main 6–9, the axes switch meanings: the input becomes each person’s share rather than the number of people, while the output becomes the number of people. Both relationships have the same formula, so students may miss this change.

The story also allows only whole-number people counts and corresponding shares such as $1,\frac12,\frac13,\ldots$. Fractional people, negative inputs, and shares larger than the whole prize do not fit the situation, although the reciprocal function accepts every nonzero real input.

A real-world context is useful when it clarifies the mathematics. Here, explaining the context’s restrictions and changing quantities may distract from the function itself. This supplement therefore introduces $y=1/x$ directly through values and its graph, allowing students to explore its behavior without first interpreting a story.

Coverage

Packet component	Support in this lesson
Main 1–5 · pp. 1–2	Reciprocal-value table, full graph, and horizontal-asymptote explanation.
Main 6–9 · pp. 2–3	The sharing context and change of axis meanings are discussed in the teacher review; inverse-function work is omitted.
Main 10–13 · pp. 3–4	Negative-input table, full graph, asymptotes, all requested features and decreasing-interval correction.
Ready 1–6 · p. 5	Linked quadratic review and all transformations/vertex forms.
Set 7–10 · p. 6	All amounts and counts, meanings, sample table; discrete plot not shown.
Go 11–18 · p. 7	All graph matches/asymptotes and both requested explanations.

Data and brief prompts adapted from Mathematics Vision Project, Secondary Mathematics III, Module 4 student packet, Section 4.1, printed pp. 1–7, under CC BY 4.0. Explanations, solutions, and diagrams were prepared for this supplement.