Understand intercepts and starting values

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Difficulty: Beginner
Estimated time: 28 minutes
Domains: Algebra
Techniques: X-interceptsY-interceptsZero-coordinate-ruleContextual-validity

What you’ll learn

Find x- and y-intercepts from equations, graphs and tables.
Connect a y-intercept with an input of zero, and an x-intercept with an output of zero.
Give an intercept as the coordinate or point the question asks for, with the right units.
Decide whether an axis crossing is a real event in the story.

Prerequisites

Why this matters on the SAT

Read an axis crossing as a real event

SAT questions rarely stop at "find the intercept." They ask what it means: a starting amount, a fixed fee, the moment something runs out. What each variable measures tells you whether that moment comes when the input is zero or when the output is zero. Either way, the key fact is the same: at an intercept, one of the two coordinates is zero.

SAT example

A reservoir contains $V$ liters of water $t$ minutes after draining begins. The relationship is modeled by

V=420-12t.

The graph of this equation in the $tV$ -plane is a line. Which choice is the best interpretation of the $t$ -intercept of the graph?

A
The reservoir initially contains $420$ liters of water.
B
The reservoir loses $12$ liters of water each minute.
C
The reservoir will be empty $35$ minutes after draining begins.
D
The reservoir contains $35$ liters of water when draining begins.

Solution to the example

The $t$ -intercept is where the graph meets the $t$ -axis, so the other quantity, $V$ , is zero. Set $V=0$ :

\begin{aligned} 0&=420-12t\\[1.4em] 12t&=420\\[1.4em] t&=35. \end{aligned}

That gives the point $(35,0)$ : after $35$ minutes, $0$ liters are left. The reservoir is empty $35$ minutes after draining begins, so the answer is C.

The other choices describe different things. Choice A, the $420$ liters at the start, is the $V$ -intercept, where $t=0$ . Choice B, $12$ liters a minute, is the rate of change you worked with in Find slope and rate of change. Choice D treats $35$ minutes as $35$ liters.

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Here x stands for t and y stands for V. The line crosses the vertical axis at the starting amount and the horizontal axis at the moment the reservoir is empty.

The zero coordinate names the intercept

An intercept is a point where a graph meets an axis.

An x-intercept sits on the x-axis, so its y-coordinate is zero. It looks like $(a,0)$ .
A y-intercept sits on the y-axis, so its x-coordinate is zero. It looks like $(0,b)$ .

So the zero tells you which axis the point is on, and the other coordinate tells you where along that axis. If a graph passes through $(-3,0)$ , that's an x-intercept, because its y-coordinate is zero. If it passes through $(0,6)$ , that's a y-intercept, because its x-coordinate is zero.

Check your understanding:

A line crosses the axes at $(0,-5)$ and $(8,0)$ . Which point is the x-intercept, and which is the y-intercept? If a question asks only for the x-coordinate of the x-intercept, what do you enter?

Common mistake:

It’s easy to call $(7,0)$ a y-intercept, because the $7$ is the number that catches your eye. But the axis comes from the zero, not from the other number. Here $y=0$ , so the point sits on the x-axis, and $(7,0)$ is an x-intercept. If you’re not sure, sketch quick axes and plot the point.

Find both intercepts from an equation or graph

Here's the one move to remember: to find where a line meets one axis, set the other variable to zero.

Take

2x+3y=12.

For the x-intercept, set $y=0$ :

\begin{aligned} 2x+3(0)&=12\\[1.4em] 2x&=12\\[1.4em] x&=6. \end{aligned}

The x-intercept is $(6,0)$ .

For the y-intercept, set $x=0$ :

\begin{aligned} 2(0)+3y&=12\\[1.4em] 3y&=12\\[1.4em] y&=4. \end{aligned}

The y-intercept is $(0,4)$ .

The graph shows the same two points: the line crosses the x-axis at $(6,0)$ and the y-axis at $(0,4)$ . With numbers this clean, working it out by hand is quicker than opening a graph.

Graph-and-click wins in two cases: the graph is already given, or the numbers are messy and a decimal answer will do. Graph the equation, click the line, then click where it crosses the axis you want. Write down the whole point, then enter the coordinate or point the question asks for.

One catch: Desmos may round the label. If the question needs an exact value, set the other variable to zero and solve by hand, and use the graph as a check. For practice with the clicks, see Read points of interest from a graph.

Check your understanding:

Find both intercepts of $4x-5y=20$ . Which variable do you set to zero for each one?

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Both marked points match what you found by setting the other variable to zero.

Read a y-intercept from slope-intercept form

When an equation is already in the form

y=mx+b,

setting $x=0$ leaves $y=b$ . So the y-intercept is $(0,b)$ , and you can read it straight off the equation.

That shortcut only works once $y$ is alone on one side. In

3y=6x-15,

the $-15$ is not the y-coordinate of the intercept, because $y$ isn't alone yet. Divide every term by $3$ :

y=2x-5.

Now you can read it: the y-intercept is $(0,-5)$ . Setting $x=0$ in the original equation gives the same point, since $3y=-15$ means $y=-5$ .

Common mistake:

Picking $-15$ as the y-intercept of $3y=6x-15$ means the constant was read before $y$ was on its own. Divide the whole equation by $3$ first, or skip the rearranging and set $x=0$ . Both give $y=-5$ , so the intercept is $(0,-5)$ .

Read intercepts in tables and stories

In a two-column table, every row is a point. So look for a zero in either column, then read the whole row.

Values of a linear function

$x$	$f(x)$
$-2$	$12$
$0$	$8$
$2$	$4$
$4$	$0$
$6$	$-4$

The row with $x=0$ gives $(0,8)$ , the y-intercept. The row with $f(x)=0$ gives $(4,0)$ , the x-intercept. You don't need a graph here: reading the zero rows is faster than typing the table into Desmos.

Now give the rows a story. Say $x$ is time in hours and $f(x)$ is the amount of liquid in a container, in liters.

$(0,8)$ means there are $8$ liters at time $0$ , so $8$ liters is the starting amount.
$(4,0)$ means there are $0$ liters after $4$ hours, so the container is empty at that point.

Each unit belongs to its own coordinate, not to the pair as a whole. Read $(4,0)$ as " $4$ hours, $0$ liters."

Common mistake:

The left edge of a graph isn’t always the y-axis. The y-intercept is only where $x=0$ , so read the axis labels before you read a crossing. Watch for this when a graph starts at a positive x-value or shows a cut-off window.

Does the intercept belong in the story?

Every intercept has a spot on the graph. But that spot doesn't always match something that can happen in the story.

In a word problem, work in this order:

Say what the input and the output measure.
Decide which one is zero at the intercept the question asks about.
Give the other coordinate with its units, keeping the point in (input, output) order.
Check that the input is allowed by the story or the stated interval.

Here's where step 4 matters. A platform starts rising, and its height, in meters, $t$ seconds later is

h(t)=1.5t+6,\qquad 0\le t\le 10.

The y-intercept is $(0,6)$ , so the platform is $6$ meters high when timing starts.

The interval $0\le t\le10$ says the story covers only the first $10$ seconds. So its graph is just the piece of the line from $t=0$ to $t=10$ . That piece is called the restricted graph.

Now ignore the interval and keep drawing the line backward. You get the whole line the formula describes, called its unrestricted linear extension, or the extension for short. So there are two graphs to keep apart: the piece the story uses, and the full line. The full line has an x-intercept:

\begin{aligned} 0&=1.5t+6\\[1.4em] t&=-4. \end{aligned}

So $(-4,0)$ is a real point on the full line. But $-4$ seconds is outside $0\le t\le10$ , so it's not on the piece the story uses, and it tells you nothing about the platform.

The last point can fool you too. At $t=10$ the model gives its final value, but that point isn't on an axis, because neither coordinate is zero. A starting value, a final value and an intercept can sometimes be the same point, so let the coordinates decide.

Which quantity should be zero? That depends on what the question asks and what each variable measures.

Set the input to zero when

The question asks for the y-intercept, $f(0)$ or where the graph crosses the vertical axis, like the point $(0,8)$ in the table.
The input counts time, items or distance from a start, so input $0$ is that start, like $t=0$ when the platform starts rising.
The question asks for a starting or base amount that comes at input $0$ , like the $420$ liters in the reservoir when draining begins.

Set the output to zero when

The question asks for the x-intercept or where the graph crosses the horizontal axis, like the reservoir’s $t$ -intercept.
The output is an amount left, so "empty," "used up" or "sold out" means the output is $0$ , like $V=0$ when the reservoir runs dry.
The event itself means zero output, like a height of $0$ at ground level or a profit of $0$ at break-even.

Whichever quantity is zero, the other coordinate tells the story, in its own units.

Input $0$ only counts if the story allows it. A parking fee that starts at $h=1$ has its first value at $h=1$ , and that isn’t a y-intercept, because the input isn’t $0$ . If the question asks about some other moment, use the input the problem gives you.

Example: What both intercepts mean

Worked example

A school club has $N(h)$ raffle tickets remaining $h$ hours after ticket sales begin. For $0\le h\le8$ , the number of tickets remaining is modeled by

N(h)=240-30h.

Which choice correctly interprets both intercepts of the graph?

A
The club begins with $8$ tickets and sells all $240$ tickets after $0$ hours.
B
The club begins with $240$ tickets and has no tickets remaining after $8$ hours.
C
The club begins with $30$ tickets and sells $240$ tickets each hour.
D
The graph has a y-intercept of $(240,0)$ and an x-intercept of $(0,8)$ .

Step 1

Say what each variable measures

The input $h$ counts hours since sales began. The output $N(h)$ counts tickets left. So each zero has its own meaning:

$h=0$ is the moment sales begin.
$N(h)=0$ means no tickets are left.

Step 2

Find the y-intercept

At the y-intercept, the input is zero:

\begin{aligned} N(0)&=240-30(0)\\[1.4em] &=240. \end{aligned}

The y-intercept is $(0,240)$ , so the club has $240$ tickets when sales begin.

Step 3

Find the x-intercept and check it

At the x-intercept, the output is zero:

\begin{aligned} 0&=240-30h\\[1.4em] 30h&=240\\[1.4em] h&=8. \end{aligned}

The x-intercept is $(8,0)$ . Is $h=8$ allowed? Yes, it's inside $0\le h\le8$ , so this point is part of the story: no tickets are left after $8$ hours.

Step 4

Match the whole meaning

Choice B gets both meanings right, with each number in its place. The y-intercept is the starting number of tickets, and the x-intercept is the time when the count reaches zero. Choice D has the right numbers in the wrong order: $240$ tickets goes with $h=0$ , so the y-intercept is $(0,240)$ , not $(240,0)$ .

Check your understanding:

Suppose the same equation only covered $0\le h\le6$ . Would $(8,0)$ still be the x-intercept of the extended line? Would it describe the tickets during the modeled time?

Practice problems

Your turn. In each problem, find the quantity that's zero before you calculate, and give your answer in the form and units the question asks for.

Let the graph handle messy decimals

Practice problem

The graph of

7.5x+4.2y=31.8

has an x-intercept at $(a,0)$ . What is the value of $a$ ?

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Graph the equation as written and click its x-axis crossing.

Read a zero row in a table

Practice problem

A tank is drained at a constant rate. The table shows the amount of water $W$ , in liters, remaining $t$ minutes after draining begins.

$t$ (minutes)	$W$ (liters)
$0$	$72$
$3$	$48$
$6$	$24$
$9$	$0$

Which choice is the best interpretation of the point $(9,0)$ ?

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The answer is in the table. If you want to see it, enter the table here and check that

(9,0)

sits on the horizontal axis.

Is the first charge a starting value?

Practice problem

The cost $C(h)$ , in dollars, to park for $h$ hours is modeled by

C(h)=9+4(h-1),\qquad 1\le h\le10.

Which choice correctly compares the restricted graph of $C$ with the unrestricted linear extension of its formula?

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Start with the allowed hours. To see the two graphs, graph

y=9+4(x-1)

, then add the restriction

1\le x\le10

and compare them at the y-axis.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

An x-intercept looks like $(a,0)$ , because the output is zero.
A y-intercept looks like $(0,b)$ , because the input is zero. For $y=f(x)$ , it's $(0,f(0))$ , as long as $0$ is an allowed input.
From an equation, set the other variable to zero and solve by hand when the numbers are clean. With messy numbers, graph and click if a decimal will do, and solve by hand if you need an exact value.
In a table, read the row with the zero.
Read each coordinate with its own variable and units: $(4,0)$ means " $4$ hours, $0$ liters."
An intercept belongs in the story only if its input is allowed and the zero event makes sense. The platform line hits $(-4,0)$ , but only on the full line, outside $0\le t\le10$ .
A graph limited to allowed inputs can miss an axis entirely: the parking graph starts at $(1,9)$ , so it has no y-intercept.
A first or last point isn't an intercept unless one of its coordinates is zero.

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Write and match linear functions

Use points, rates, intercepts, tables, and graphs to identify or build a complete linear function.

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