Sliders for unknown constants

Your feedback matters:

Lesson progressPractice problems 0/4

Difficulty: Intermediate
Estimated time: 35 minutes
Domains: Algebra Advanced Math
Techniques: SlidersConstants

What you’ll learn

Recognize an unknown constant that Desmos can control with a slider.
Create one slider and set useful bounds or a step size.
Match a graph to a stated point or intersection condition.
Type exact candidate values instead of trusting a visual estimate.
Apply sign, range, and integer restrictions before choosing a value.
Calculate the quantity the question asks for after finding the constant.
Choose algebra when it is shorter or more exact.

Prerequisites

Why this matters on the SAT

Match graph conditions without rearranging the formula

SAT questions often include a letter described as a constant. That letter may control where a graph sits or where two graphs intersect. A Desmos slider lets you vary one candidate value while keeping the rest of the problem visible.

SAT example

The function $f$ is defined by

f(x)=a(x-4)^2-3,

where $a$ is a constant. The graph of $f$ passes through the point $(2,9)$ . What is the value of $f(0)$ ?

A
$3$
B
$9$
C
$45$
D
$48$

Fast Desmos solution

Enter f(x)=a(x-4)^2-3, add a slider for $a$ , and plot $(2,9)$ . Set $a=3$ so that the curve passes through the point. Then enter f(0). Desmos returns $45$ , so the answer is C.

The slider finds the constant, but the problem asks for a function value. Always finish with the exact quantity requested.

Calculator loads as you approach

At a = 3, the curve passes through (2, 9) and f(0) = 45.

When should you use a slider?

A slider is strongest when one unknown constant controls a visible graph condition and testing exact values is clearer than rearranging the formula.

Begin with a slider when…

one unknown constant changes a graph or function;
the target is visible, such as passing through a point or intersecting at a stated coordinate;
answer choices or a restriction give a small set of sensible values;
moving the constant helps you understand what it controls; or
the calculator setup is clearer than the algebra and reasonably short.

Use algebra when…

one substitution isolates the constant immediately;
the exact answer is an awkward fraction that a slider may only approximate; or
several unknown constants and conditions call for a later regression workflow.

Look for wording such as “where $k$ is a constant,” “the graph passes through the point,” “the solution has an $x$ -coordinate of,” “which value of $c$ could make the statement true?”, or “ $a$ is a positive integer.” Type exact candidate values, then verify the final value in the original condition. Questions about exactly zero, one, or many solutions need the later solution-count workflow.

Check your understanding:

A line $y=3x+b$ passes through $(2,8)$ . A different problem asks which value of $c$ from $1$ , $3$ , $5$ , or $7$ makes $g(x)=|x-c|+\frac{x^2}{4}$ pass through $(6,12)$ . Which problem is more likely to benefit from a slider, and why?

1. Create one slider and see what changes

In Desmos, lowercase $x$ and $y$ are graphing variables. A different undefined letter, such as $a$ , $b$ , $c$ , or $k$ , can represent a constant.

Enter

y=2x+c.

Desmos offers to add a slider for $c$ . You can also define it on another line by entering c=0.

Slider settings can make the search more useful:

change the minimum and maximum to match a stated range;
use a step of $1$ when the constant must be an integer;
use a smaller step to explore a simple decimal; and
type a value directly when you need an exact candidate.

Select the slider’s limits control beside the expression to reveal its minimum, maximum, and Step fields.

Try it yourself:

Set the minimum to $-4$ , the maximum to $5$ , and the step to $1$ . Move $c$ through the allowed values and watch the $y$ -intercept. What stays unchanged?

The slider now visits only integers from $-4$ through $5$ . The slope remains $2$ while the vertical position and $y$ -intercept change.

Check your understanding:

Why is a step of $1$ more useful than a step of $0.1$ when $c$ must be an integer? At $c=-4$ , what are the line’s slope and $y$ -intercept?

Common mistake:

Using $x$ as the slider letter or defining the same constant on two lines. Keep $x$ as the horizontal variable and define the slider constant only once.

Calculator loads as you approach

Changing c moves the line vertically while its slope stays 2.

2. Match the condition, then finish the question

A point $(u,v)$ lies on a function’s graph exactly when the function returns $v$ at input $u$ .

Suppose

p(x)=b(x+1)^2-4

passes through $(2,23)$ . What is $p(0)$ ?

Enter the function, add a slider for $b$ , and plot the target point.

Use this workflow:

Keep the function and target point visible.
Move the slider to understand the direction of change.
Type exact candidate values into the slider.
Verify the value in the point condition.
Enter the expression the question actually asks for.

Check your understanding:

Use the graph and point condition above. What are the values of $b$ and $p(0)$ ?

Common mistake:

Submitting the slider value when the question asks for an output, coordinate, or expression involving the constant.

Calculator loads as you approach

Adjust b until the graph contains (2, 23), then evaluate the requested output.

3. Example: Match a system to a stated coordinate

A slider can also control one equation in a system. Keep both equations visible, then adjust the constant until their intersection matches the stated coordinate.

Worked example

The graphs of

y=2x+5

and

y=kx-1,

where $k$ is a constant, intersect at a point that

has an $x$ -coordinate of $3$ .

What is the value of

$2k$ ?

A
$4$
B
$6$
C
$8$
D
$10$

Step 1

Identify the target

The two graphs must intersect where $x=3$ . The point’s $y$ -coordinate is not given, so the intersection itself will supply it.

Step 2

Graph both equations

Enter both equations and add a slider for $k$ . Keep the graph wide enough to see the intersection.

Calculator loads as you approach

Adjust k until the intersection has x = 3.

Step 3

Set and verify the constant

At $k=4$ , the graphs intersect at $(3,11)$ . Verify the point in both equations:

2(3)+5=11

and

4(3)-1=11.

The same point satisfies both equations, so $k=4$ is exact.

Step 4

Answer what was asked

The question asks for $2k$ , not $k$ :

2k=2(4)=8.

The answer is C.

Common mistake:

Stopping at $k=4$ . A slider often finds an intermediate value, so reread the final question before submitting.

4. Apply every restriction

A graph may satisfy the visible condition for more than one slider value. A sign or range restriction can decide which value is valid.

Consider

g(x)=|x-c|+2,

where $c>0$ . The graph passes through $(1,7)$ . What is the value of $c$ ?

The point condition gives

|1-c|+2=7,

so both $c=-4$ and $c=6$ make the graph pass through $(1,7)$ . The restriction $c>0$ leaves

c=6.

When a constant must be positive, negative, integral, or inside an interval, set useful slider bounds or keep the restriction beside the graph. Never accept a matching value that violates the problem statement.

Check your understanding:

Why is $c=-4$ not an answer even though its graph passes through $(1,7)$ ?

Common mistake:

Stopping at the first slider value that works. The condition may allow another value, and the stated restriction may select the answer.

Calculator loads as you approach

Both c = -4 and c = 6 match the point, but only one satisfies c > 0.

Finish the solution

The function, target point, and slider are already entered. Match the point, verify the constant, evaluate the requested output, and submit that output.

Match the point and evaluate

Finish the solution

The function $q$ is defined by

q(x)=d(x-1)^2+2,

where $1<d<4$ . The graph of $q$ passes through the point $(4,24.5)$ . What is the value of $q(-1)$ ?

First steps

The function, point, and $d$ slider are already visible.
Set the slider minimum to $1.5$ , maximum to $3.5$ , and step to $0.5$ .
Type candidate values until the curve passes through $(4,24.5)$ .
Verify that $q(4)=24.5$ .
Enter q(-1) and submit its value.

Finish it

Your answer

Calculator loads as you approach

Use the stated range and a 0.5 step to match (4, 24.5), then evaluate q(-1).

Practice problems

SAT practice problems

Use one slider only when it clarifies the constant’s effect or makes exact candidates easy to test. Keep the stated condition visible, type exact values, verify the winner, and finish the requested calculation.

Choose the shorter method

Practice problem

The function $f$ is defined by

f(x)=mx-4,

where $m$ is a constant. The graph of $f$ passes through $(5,11)$ . What is the value of $f(2)$ ?

Calculator loads as you approach

Match a system coordinate

Practice problem

In the system below, $a$ is a constant:

\begin{aligned} 2x-3y&=1\\ ax+6y&=2a. \end{aligned}

The system has a solution with $x=1$ . What is the value of $a$ ?

Calculator loads as you approach

Apply a restriction

Practice problem

The function $g$ is defined by

g(x)=(x-c)^2+2,

where $c>1$ . The graph of $g$ passes through $(1,27)$ . What is the value of $c+2$ ?

Calculator loads as you approach

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Use a different letter from $x$ and $y$ for an unknown constant.
Move a slider to understand the graph, then type exact candidate values.
Match every stated point or intersection condition.
Set bounds and steps that respect sign, range, or integer restrictions.
Verify the final constant in the original equation or function.
Calculate the output, coordinate, or expression the question actually asks for.
Choose algebra for short exact work and avoid visual guesses with awkward fractions.

Next lesson

How many solutions?

Use intersections, tangency, and graph behavior to determine when a parameter creates zero, one, or multiple solutions.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Match graph conditions without rearranging the formula

Fast Desmos solution

When should you use a slider?

Begin with a slider when…

Use algebra when…

1. Create one slider and see what changes

2. Match the condition, then finish the question

3. Example: Match a system to a stated coordinate

Identify the target

Graph both equations

Set and verify the constant

Answer what was asked

4. Apply every restriction

Finish the solution

Match the point and evaluate

First steps

Finish it

Practice problems

SAT practice problems

Choose the shorter method

Match a system coordinate

Apply a restriction

4 practice examples left

Quick recap

How many solutions?

Free question bank practice problems