Custom regression with multiple conditions

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Difficulty: Advanced
Estimated time: 45 minutes
Domains: Algebra Advanced Math
Techniques: RegressionCustom-regressionMatched-listsConstraintsFunctions

What you’ll learn

Count unknown parameters and independent conditions before fitting.
Distinguish independent information from repeated information.
Combine mixed conditions in equal-length, positionally matched lists.
Build a table when every condition is an input-output point.
Preserve the exact given model and keep only the intended parameters free.
Read fitted values as numerical candidates and recover exact values when justified.
Answer the requested quantity and verify every original condition.
Choose substitution or ordinary system graphing when either is shorter.

Prerequisites

Custom regression for one unknown parameter0%
Lesson 17: preserve a given model, fit one unknown parameter from one condition, and verify the result.

Why this matters on the SAT

Fit several exact conditions without dismantling the model

Some SAT questions give a function with several unknown parameters and enough exact conditions to determine them. This lesson extends custom regression for one unknown parameter: instead of fitting one parameter from one condition, you will fit several parameters from several independent conditions.

See the method on an SAT-style question.

SAT example

The quadratic function $q$ is defined by

q(x)=a(x-1)^2+b,

where $a$ and $b$ are constants. If $q(0)=12$ and $q(4)=0$ , what is the maximum value of $q$ ?

Fast Desmos solution

Put both outputs and both model values into matching lists on one regression line:

[12,0]~[a(0-1)^2+b,a(4-1)^2+b]

Desmos reports $a=-1.5=-\frac32$ and $b=13.5=\frac{27}{2}$ . Since the model is in vertex form and $a<0$ , the maximum occurs at $x=1$ and equals

q(1)=b=\boxed{13.5}.

The list does not change the model. It simply tells Desmos that the same values of $a$ and $b$ must satisfy both conditions.

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Two exact function values determine a and b, and q(1) gives the maximum.

When should you use this method?

Use multiple-condition custom regression when:

the problem gives a specific model;
the model has several unknown parameters;
the problem supplies at least as many independent exact conditions as unknown parameters;
the same parameter values must satisfy every condition.

Choose the setup that matches the information:

Use a custom regression when…

Table custom regression: every condition is a point or function value, such as $f(1)=7$ and $f(3)=31$ .
Matched-list custom regression: the conditions have different forms, such as a function value and a relationship between parameters.

Choose a shorter method when…

Substitution: one condition immediately isolates a parameter or makes the remaining equation simple.
Ordinary system graphing: the problem is simply a system in $x$ and $y$ . Enter the equations with equals signs and read their intersection.

Regression from a supplied table uses an automatic model to summarize tabular data. Here, the SAT gives the exact model and exact conditions, so you preserve that custom form. This lesson also differs from parameterized points and answer choices, where a parameter moves a point through infinitely many positions instead of being fitted from conditions.

The core examples here are linear in the fitted parameters. A parameter may multiply a known expression or be added to another term, but it does not appear in an exponent, denominator, root, or product with another fitted parameter. Noisy data, numerical-search behavior, identities sampled across many inputs, tangency, circles, percent chains, and regression failure modes belong in later lessons. A later lesson on regression failure modes will handle cases where the output cannot be treated as an exact solution.

Check your understanding:

You need to solve $2x+y=11$ and $x-y=1$ . Should you begin with matched-list regression or ordinary system graphing?

1. Count parameters and independent conditions

Start by counting the undefined constants that Desmos must fit. Then count the conditions that give genuinely new information.

As a working rule:

\text{number of independent conditions} \ge \text{number of unknown parameters}.

Having the same number is not enough if two conditions repeat the same information.

Consider

f(x)=a(x-2)^2+b

with

f(0)=7 \quad\text{and}\quad f(4)=7.

The model has two unknown parameters, $a$ and $b$ . The first condition gives

4a+b=7.

The second condition also gives

4a+b=7.

These conditions are redundant. They describe symmetric points on the same horizontal level, but they provide only one independent equation. Infinitely many pairs $(a,b)$ satisfy that equation, so the function is not determined.

An additional condition such as $f(2)=3$ would produce

b=3,

which is independent of $4a+b=7$ . Together, the two independent equations would determine both parameters.

Check your understanding:

For $h(x)=mx+c$ , the conditions are $h(0)=4$ , the $y$ -intercept is $4$ , and $h(3)=10$ . How many independent conditions are present, and are they enough to determine $m$ and $c$ ?

2. Put mixed conditions on one matched-list line

When the conditions are not all input-output points, place corresponding sides into two equal-length lists:

[known value 1,known value 2]~[model expression 1,model expression 2]

Desmos matches entries by position:

the first left entry pairs with the first right entry;
the second left entry pairs with the second right entry;
every pair uses one shared set of fitted parameter values.

The two lists must have equal lengths. Keep the conditions in a clear order so each known value remains paired with its own model expression.

In the opening example, the line

[12,0]~[a(0-1)^2+b,a(4-1)^2+b]

means

\begin{aligned} 12&=a(0-1)^2+b,\\ 0&=a(4-1)^2+b. \end{aligned}

Both equations are fitted together. This is why Desmos finds one shared pair, $a=-\frac32$ and $b=\frac{27}{2}$ .

Common mistake:

Putting the conditions on separate regression lines. Separate regression lines are separate fits. They do not reliably enforce one shared set of parameter values. Combine all required conditions on one matched-list line.

Check your understanding:

Translate $2a+b=9$ and $a-3b=-5$ into one matched-list regression line. Then state what the list order means.

After you translate the pair, use the calculator to confirm that one matched-list regression fits one shared value of $a$ and one shared value of $b$ . Edit either list entry to see how both fitted values respond, then reset the authored state.

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The matched lists fit one shared pair of values for a and b.

3. Use a table when every condition is a point

Worked example

The function $f$ is defined by

f(x)=a\cdot2^x+b,

where $a$ and $b$ are constants. If $f(1)=7$ and $f(3)=31$ , what is the value of $b$ ?

Both conditions are function values, so they are the points $(1,7)$ and $(3,31)$ . Put those points in a table, then enter the exact custom model:

y_1~a*2^(x_1)+b

The conditions produce

\begin{aligned} 2a+b&=7,\\ 8a+b&=31. \end{aligned}

Subtracting gives $6a=24$ , so $a=4$ and $b=-1$ . The requested value is

\boxed{-1}.

This is not automatic exponential regression. The base $2$ is fixed, and the vertical shift $b$ is part of the given model. An automatic exponential template can introduce a different set of free parameters. Custom regression preserves exactly

a\cdot2^x+b.

Try it yourself:

Change the second table input from $3$ to $2$ and its output from $31$ to $15$ . Predict whether $a=4$ and $b=-1$ will still satisfy both points, then reset the example.

Check your understanding:

Choose a table or matched lists for each setup: (1) $p(2)=5$ and $p(6)=17$ for a stated model; (2) $p(2)=5$ and the parameters satisfy $a+b=4$ .

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The two table rows determine a = 4 and b = -1 in the exact fixed-base model.

4. Keep only the intended parameters free

Before fitting, inspect every undefined letter.

For

f(x)=a\cdot2^x+b,

the intended fitted parameters are $a$ and $b$ . The base $2$ is fixed by the problem. In a table regression:

x_1 and y_1 are the paired data lists;
$a$ and $b$ remain undefined so Desmos can fit them;
every other number and operation stays exactly as given.

Do not replace the fixed base with another free letter. A model such as

y_1~a*c^(x_1)+b

has three unknown parameters, $a$ , $b$ , and $c$ , but only two point conditions. It no longer represents the given problem.

Also check that neither intended parameter was defined on an earlier expression line. If $a$ or $b$ already has a value, Desmos treats it as fixed instead of fitting it. Delete the old definition or use a fresh parameter name consistently.

Exact conditions still need exact verification

Regression output is numerical. A displayed value such as $-1.5$ suggests $-\frac32$ , but the fraction is exact only after it satisfies the original equations exactly.

For the opening example:

\begin{aligned} q(0) &=-\frac32(0-1)^2+\frac{27}{2} =12,\\ q(4) &=-\frac32(4-1)^2+\frac{27}{2} =0. \end{aligned}

Both original conditions are true, so the exact values are confirmed.

An extra exact condition can be useful as a check. It does not replace the need for enough independent conditions. Three copies of $4a+b=7$ still provide only one independent equation.

If even one required exact condition fails, reject the fitted values and correct the setup before answering. A result that satisfies only some of the conditions is not a solution to the stated problem.

5. Example: Match mixed conditions

Worked example

The function $f$ is defined by

f(x)=ax+b,

where $a$ and $b$ are constants. The function $g$ is defined by

g(x)=f(2x).

If $g(3)=5$ and $a+b=0$ ,

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

A. $-1$

B. $0$

C. $1$

D. $2$

Step 1

Translate the transformed condition

The condition $g(3)=5$ belongs to $g$ , not $f$ . Substitute $2x$ into the definition of $f$ :

g(x)=f(2x)=a(2x)+b.

Therefore,

g(3)=a(2\cdot3)+b=6a+b=5.

The second condition, $a+b=0$ , is already a relationship between the parameters.

Step 2

Match the mixed conditions by position

One condition is a function value and the other is a parameter relationship, so use matched lists rather than a table:

[5,0]~[a(2*3)+b,a+b]

Desmos reports $a=1$ and $b=-1$ .

Calculator loads as you approach

The first list pair encodes g(3) = 5, and the second encodes a + b = 0.

Step 3

Answer the requested quantity

The regression fitted $a$ and $b$ , but the question asks for $f(0)$ . Return to the original function:

f(0)=a(0)+b=b=-1.

The answer is A.

Step 4

Verify every original condition

With $a=1$ and $b=-1$ ,

g(x)=1(2x)-1=2x-1.

Now check both conditions:

\begin{aligned} g(3)&=2(3)-1=5,\\ a+b&=1+(-1)=0. \end{aligned}

Both original conditions hold, and $f(0)=-1$ answers the requested quantity.

Common mistake:

Trying to put both conditions into a table. The condition $a+b=0$ is not an input-output point, so keep it as the second expression in the matched lists.

6. Read the output, answer, and verify

A complete multiple-condition regression solution has three separate jobs:

Read the fitted parameters. Treat the displayed decimals as candidates for exact values.
Answer the requested quantity. The SAT may ask for a function value, intercept, maximum, or expression rather than a fitted parameter.
Verify every condition. Substitute the fitted values into each original condition, not only one of them. If any required exact condition fails, stop and fix the setup.

For the guided example, $a=1$ and $b=-1$ are fitted values, $f(0)=-1$ is the requested answer, and the calculations $g(3)=5$ and $a+b=0$ verify the setup.

Finish the solution

The matched-list regression is already entered. Use the fitted parameters to evaluate the requested function value.

Evaluate the quadratic function

Finish the solution

The quadratic function $q$ is defined by

q(x)=a(x-3)^2+b,

where $a$ and $b$ are constants. If $q(0)=23$ and $2a-b=-1$ , what is the value of $q(5)$ ?

First steps

Translate $q(0)=23$ as $a(0-3)^2+b=23$ .
Keep the parameter condition $2a-b=-1$ .
Fit both conditions together with [23,-1]~[a(0-3)^2+b,2a-b].

Finish it

Your answer

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The shared regression has fitted a and b. Add q(5) to finish.

Practice problems

SAT practice problems

For each problem, count the unknown parameters and independent conditions first. Then decide whether a table, matched lists, substitution, or ordinary system graphing is shortest. If you use regression, verify every condition before accepting the requested answer.

Combine a point and a parameter relationship

Practice problem

The linear function $f$ is defined by

f(x)=ax+b,

where $a$ and $b$ are constants. If $f(5)=15$ , the slope of the graph of $f$ is $3$ less than its $y$ -intercept, and $f(0)=5$ , what is the value of $f(-2)$ ?

Your answer

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Translate the function value and slope-intercept relationship into one matched-list regression.

Fit a shifted linear cost model

Practice problem

The function $C$ models the cost, in dollars, of processing $g$ grams of a material. The model is

C(g)=a+b(g-5),

where $a$ and $b$ are constants. The graph of $C$ passes through $(3,52)$ and $(9,88)$ . What is the value of $a$ ?

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Enter the two cost points in a table and fit the exact shifted linear model.

Preserve a two-parameter quadratic model

Practice problem

The quadratic function $f$ is defined by

f(x)=a(x-2)^2+b,

where $a$ and $b$ are constants. If $f(1)=7$ and $f(5)=31$ , what is the value of $f(8)$ ?

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Use the two function values as table points, preserve vertex form, and evaluate f(8).

Choose the shortest method

Practice problem

The quadratic function $f$ is defined by

f(x)=a(x-4)^2+b,

where $a$ and $b$ are constants. If $f(4)=7$ and $f(6)=19$ , which approach is shortest for determining $f(8)$ ?

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Choose the shortest method before typing, then use the calculator to verify f(8).

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Recognize: Look for a specific model with several unknown parameters and enough exact independent conditions.
Choose: Use a table when every condition is a point. Use one matched-list line when condition types differ. Prefer substitution or ordinary system graphing when either is shorter.
Fit: Preserve the exact model, pair equal-length lists by position, and keep only the intended parameters free.
Verify: Read numerical outputs as candidates, answer the requested quantity, and check every original condition exactly.

Next lesson

Parameterized points and answer choices

Graph how a point moves as its parameter changes and test which answer choice satisfies a linear system.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Fit several exact conditions without dismantling the model

Fast Desmos solution

When should you use this method?

Use a custom regression when…

Choose a shorter method when…

1. Count parameters and independent conditions

2. Put mixed conditions on one matched-list line

3. Use a table when every condition is a point

4. Keep only the intended parameters free

Exact conditions still need exact verification

5. Example: Match mixed conditions

Translate the transformed condition

Match the mixed conditions by position

Answer the requested quantity

Verify every original condition

6. Read the output, answer, and verify

Finish the solution

Evaluate the quadratic function

First steps

Finish it

Practice problems

SAT practice problems

Combine a point and a parameter relationship

Fit a shifted linear cost model

Preserve a two-parameter quadratic model

Choose the shortest method

5 practice examples left

Quick recap

Parameterized points and answer choices

Free question bank practice problems