Custom regression for one unknown parameter

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Difficulty: Advanced
Estimated time: 40 minutes
Domains: Algebra Advanced Math
Techniques: RegressionCustom-regressionConstantsFunctionsModel-translation

What you’ll learn

Recognize a function model with one unknown parameter and one independent condition.
Translate a function value, graph point, intercept, or known extremum into a regression condition.
Choose between direct condition regression and a table-based custom regression.
Enter a custom regression with the tilde ∼.
Preserve the exact model instead of introducing extra parameters.
Reuse the fitted parameter to answer the quantity the question asks for.
Verify a numerical regression result with the original condition.
Choose direct substitution when it is clearly shorter.

Prerequisites

Why this matters on the SAT

Keep the model intact and solve only the missing constant

The SAT can give you a function whose form is already known, leave one parameter unknown, and provide one condition that determines it. A parameter is a constant that controls part of a model, such as its slope, vertical stretch, or shift.

Custom regression lets you enter the given form exactly. Instead of expanding, rearranging, and solving by hand, you write the condition with a tilde and let Desmos recover the one missing parameter.

See the advantage on a typical SAT question.

SAT example

The linear function $f$ is defined by

f(x)=(p-3)x+(2p+4),

where $p$ is a constant. If $f(4)=16$ , what is the value of $f(9)$ ?

Fast Desmos solution

Substitute $4$ and $16$ into the given model, but use a tilde: 16~(p-3)*4+(2p+4). Desmos finds $p=4$ . Define the function and evaluate f(9) to get $\boxed{21}$ .

For a simple model, substitution may be just as fast. Custom regression is most useful when the parameter appears more than once, the model is nested or transformed, or keeping the original structure avoids messy algebra.

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One numerical condition determines p, and the fitted value is reused to evaluate f(9).

When should you use this method?

Look for all four signals:

The problem gives a specific function form.
Exactly one parameter in that form is unknown.
One independent numerical condition tells you something about the function.
For the deterministic method taught here, the parameter appears in first-degree terms, not as a square, root, denominator, or exponent.

Common conditions include:

$f(6)=11$ ;
the graph passes through $(6,11)$ ;
the graph has an $x$ -intercept at $x=6$ ;
the graph has a $y$ -intercept of $11$ ;
the model’s maximum or minimum occurs at a known point.

One unknown parameter needs one independent condition. If the model still contains two or more unknown constants, one point usually cannot determine them. A later lesson on custom regression with multiple conditions handles that broader setup.

Here, independent means the condition actually gives new information about the parameter. If

q(x)=a(x-2)^2-5,

then $q(2)=-5$ is true for every value of $a$ . The point $(2,-5)$ cannot determine $a$ because the parameter term becomes $0$ there.

Use direct substitution when…

the condition creates an equation you can solve in one or two easy steps.

Use a Desmos method when…

one numerical condition fits the complete model more cleanly as a direct regression;
a point, graph, or table makes the input-output pair clearer in a table; or
a slider helps explore a visual condition such as tangency or a changing number of intersections. Confirm the result exactly afterward.

Choose the shortest reliable opening. For example, if $f(x)=kx+12$ and $f(6)=0$ , then $6k+12=0$ gives $k=-2$ with almost no work. Substitution is faster than either regression setup.

Check your understanding:

The function $f(x)=3x+k$ satisfies $f(5)=19$ . Which method should you begin with: substitution, direct condition regression, a table, or a slider?

Check your understanding:

A function has the form $f(x)=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are all unknown, and the graph passes through only one known point. Is one-row custom regression enough to determine the function?

1. Turn the condition into a regression equality

A function condition gives the two sides of a custom regression. Substitute the known input into the given model, place the known output on the other side, and replace the equals sign with a tilde.

For common conditions:

$f(c)=d$ or a point $(c,d)$ gives d~ followed by the model with $c$ substituted for its input.
An $x$ -intercept at $x=c$ gives output $0$ .
A $y$ -intercept of $d$ gives input $0$ and output $d$ .
A known extremum $(c,d)$ gives input $c$ and output $d$ only when the model’s structure or separate reasoning already establishes that the extremum occurs at $x=c$ . Verify the location and maximum or minimum type separately.

Suppose

q(x)=a(x-2)^2-5

and the graph passes through $(6,11)$ . Enter:

11~a(6-2)^2-5

Common mistake:

Substituting the output as the input. In $f(c)=d$ , $c$ goes into the model’s input and $d$ is the value on the other side.

Check your understanding:

The graph of $r$ has an $x$ -intercept at $x=-3$ . How can you represent this condition in a direct regression or a table?

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The condition q(6) = 11 directly determines a, which is reused in q(0).

2. Use a table when it makes the condition clearer

A table expresses the same condition as a point. Put the input in x_1, the output in y_1, and then replace:

the model’s input variable with x_1;
the model’s output with y_1;
the equals sign with the tilde $\sim$ .

For

q(x)=a(x-2)^2-5,

enter:

y_1~a(x_1-2)^2-5

The table has one point and the model has one unknown parameter, so Desmos reports $a=1$ .

Try it yourself:

Change the table output from $11$ to $27$ . Predict how $a$ and $q(0)$ will change, check your prediction in the calculator, then reset the example.

Do not choose automatic quadratic regression here. Its standard form,

y_1\sim ax_1^2+bx_1+c,

introduces three free parameters. One point cannot determine all three. The problem has already told you the more specific model, so use that exact form.

The tilde is essential. An equals sign graphs or defines a relation; it does not ask Desmos to fit the undefined parameter.

Every model in this lesson is linear in the fitted parameter. The parameter is only multiplied by known expressions or added in first-degree terms, even when the function itself is quadratic, exponential, or rational in $x$ . Desmos can fit this kind of parameter deterministically. Custom regression can also fit parameters in exponents, denominators, roots, and other nonlinear positions, but those models require numerical search and can introduce failure modes that a later lesson will examine.

Check your understanding:

The graph of $r(x)=\frac{c}{x+1}+4$ passes through $(3,7)$ . Should you begin with a direct or table-based regression? Write the entry and state the value Desmos should find.

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The custom model preserves the known vertex and solves only for a.

3. Keep only the intended parameter free

Before running a regression, check that the only undefined letter is the parameter you want Desmos to find.

For example, in

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

$x$ is the input variable and $a$ is the free parameter. The numbers $3$ and $-2$ are fixed parts of the model.

In a table regression:

x_1 contains known inputs;
y_1 contains known outputs;
$a$ remains undefined so Desmos can fit it.

If $a$ was already defined on another expression line, Desmos treats it as fixed. Delete that definition or use a fresh parameter letter.

The parameter does not need a subscript. Either a or a_1 can work, but use the same name consistently. Only the table variables must match the actual column labels.

4. Example: Fit a composed model

Worked example

The functions $f$ and $g$ are defined by

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

and

g(x)=f(2x+1)+4,

where $a$ is a constant.

The graph of $g$ passes through $(2,22)$ .

What is the value of $f(0)$ ?

Step 1

Count the unknowns and conditions

The model has one unknown parameter, $a$ . The point $(2,22)$ gives one independent condition, so direct custom regression can determine $a$ .

Step 2

Write the complete model for g

Substitute $2x+1$ into $f$ :

g(x)=a\big((2x+1)-3\big)^2-2+4.

You can keep this nested structure. The regression must represent $g$ , because the known point lies on the graph of $g$ , not the graph of $f$ .

Step 3

Fit the one parameter

Enter the condition directly:

22~a((2*2+1)-3)^2-2+4

Desmos reports $a=5$ .

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The condition belongs to g, so the regression uses the complete composed model for g.

Step 4

Answer the requested function value

The question asks for $f(0)$ , not $a$ or $g(0)$ . Use the original definition of $f$ :

\begin{aligned} f(0) &=5(0-3)^2-2\\ &=5(9)-2\\ &=\boxed{43}. \end{aligned}

Verify the fitted value with the given condition:

g(2)=5(2\cdot2-2)^2+2=5(4)+2=22.

The parameter satisfies the condition, and the requested value is $43$ .

Common mistake:

Using the point $(2,22)$ in the model for $f$ . The point lies on $g$ , so the regression must include the composition and outside shift.

5. Read the output as a candidate, then verify

Desmos stores the fitted parameter, so define the original function and use the parameter letter in later calculations. This avoids retyping a rounded display value.

Regression output is numerical. If Desmos shows a clean decimal such as $4.5$ , the answer choices or fraction conversion can suggest the intended value $\frac92$ . Confirm it by exact substitution into the original condition. A rounded display alone does not prove an exact fraction.

For an exact SAT model, verify by substituting the candidate back into the condition:

Does the fitted parameter make the known condition true?
Did you use the correct function or transformation?
Did you answer the requested parameter, function value, intercept, or other quantity?
Did you preserve any domain condition?

Finish the solution

The direct condition regression and function definition are entered. Use the fitted parameter to evaluate the requested function value.

Evaluate the rational function

Finish the solution

The function $h$ is defined by

h(x)=\frac{k}{x^2-1},

where $k$ is a constant and $x\ne-1,1$ . If $h(2)=3$ , what is the value of

h\left(\frac12\right)?

First steps

Substitute $x=2$ and $h(2)=3$ into the model.
Enter the direct condition regression 3~k/(2^2-1).
Desmos reports $k=9$ .

Finish it

Your answer

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The regression has found k. Add h(1/2) to finish the problem.

Practice problems

SAT practice problems

For each problem, identify the one free parameter and choose the shortest reliable method. Use substitution when the equation is immediate. Use direct or table-based custom regression when preserving the full model saves work.

Choose substitution for a short model

Practice problem

The function $f$ is defined by

f(x)=a\cdot3^{x-1}-5,

where $a$ is a constant. If $f(2)=4$ , what is the value of $f(0)$ ?

Your answer

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Choose the shortest method, determine a, and evaluate f(0).

Keep both occurrences of the parameter

Practice problem

The linear function $g$ is defined by

g(x)=(p-2)x+(3p-1),

where $p$ is a constant. If $g(4)=19$ , what is the value of $g(9)$ ?

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Enter the numerical condition directly, keep both occurrences of p, and evaluate g(9).

Fit a parameter from a maximum point

Practice problem

The quadratic function $q$ is defined by

q(x)=a(x+1)(x-7),

where $a$ is a constant. The maximum value of $q$ is $8$ , and it occurs at $x=3$ . What is the value of $q(0)$ ?

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Use the stated maximum point to fit a, then evaluate q(0).

Fit a transformed rational model

Practice problem

The functions $f$ and $g$ are defined by

f(x)=\frac{a}{x-2}+1

and

g(x)=2f(x+1)-3,

where $a$ is a constant. If $g(4)=7$ , what is the value of $f(-1)$ ?

Your answer

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Build the complete model for g, fit a from g(4) = 7, and then evaluate f(-1).

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Recognize: Use this technique when a specific model has one unknown parameter and one independent numerical condition.
Choose: Prefer easy substitution when it is shorter; otherwise use a direct regression for a numerical condition or a table when a supplied point is clearer.
Fit: Preserve the exact given model, use a tilde, and keep only the intended parameter undefined.
Verify: Reuse the fitted parameter, answer the requested quantity, and confirm the result with the original condition and domain.

Next lesson

Custom regression with multiple conditions

Combine several exact conditions in one matched-list or table-based custom regression.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Keep the model intact and solve only the missing constant

Fast Desmos solution

When should you use this method?

Use direct substitution when…

Use a Desmos method when…

1. Turn the condition into a regression equality

2. Use a table when it makes the condition clearer

3. Keep only the intended parameter free

4. Example: Fit a composed model

Count the unknowns and conditions

Write the complete model for g

Fit the one parameter

Answer the requested function value

5. Read the output as a candidate, then verify

Finish the solution

Evaluate the rational function

First steps

Finish it

Practice problems

SAT practice problems

Choose substitution for a short model

Keep both occurrences of the parameter

Fit a parameter from a maximum point

Fit a transformed rational model

5 practice examples left

Quick recap

Custom regression with multiple conditions

Free question bank practice problems