Solve by Table/Function Regression

Exponential Function Decay Rate on the SAT: Desmos Walkthrough

Drop the two given points into a Desmos table, fit the exact model

y_1\sim a^{x_1}+b

with one regression line, read off

a

and

b

, and compute the value the question asks for.

This lesson uses Desmos tables and regressions: drop the given values into a table, fit the model with a regression line, and read the constants straight off the fit.

Take this lesson interactively

A real SAT-style question

The function $h$ is defined by $h (x) = a^{x} + b$ , where $a$ and $b$ are positive constants. The graph of $y = h (x)$ in the $x y$ -plane passes through the points ( $0$ , $10$ ) and ( $-2$ , $\frac{325}{36}$ ). What is the value of $a b$ ?

$\frac{1}{4}$

$\frac{1}{2}$

$54$

$60$

How to solve it in Desmos, step by step

The idea

The function is really h of x equals a to the power x plus b. Drop the two points into a table, fit that exact model with one regression line, and Desmos solves both constants at once, then multiply them.

1
Step 1
Open the add menu in the top left and choose Table. In the x sub 1 column enter 0 in the first row and negative 2 in the second row. In the y sub 1 column enter 10 in the first row and 325 divided by 36 in the second row. These are the two points the graph passes through.
Type in Desmos
$x_{1}=\left[0,-2\right]$ $y_{1}=\left[10,\frac{325}{36}\right]$
2
Step 2
In the next empty expression line type this regression for the actual function, a to the power x plus b. Desmos fits it to your two points and shows the solved values just below the line. Read off a equals 6 and b equals 9.
Type in Desmos
$y_{1}\sim a^{x_{1}}+b$
3
Step 3
In the next empty line type a times b. Desmos displays 6 times 9 equals 54 to the right. That matches choice C.
Type in Desmos
$ab$

The answer is C.

$54$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

The function

h

is defined by

h(x) = a^x + b

, where

a

and

b

are positive constants. The graph of

y = h(x)

in the

xy

-plane passes through the points

(0, 8)

and

\left(-2, \frac{113}{16}\right)

. What is the value of

ab

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson

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