Question 12·SAT Desmos Regression Practice·Advanced Math

0 / 68 Questions completed

In a beadwork design, the number of beads in row $n$ is modeled by a quadratic function $B(n)$ , where $n$ is a row number.

The design satisfies the following values:

$B(1)=10$
$B(4)=49$
$B(6)=95$

Which choice is the value of $B(3)$ ?

With three values of a quadratic function and no given vertex or roots, use a Desmos table to fit a rule of the form $an^2+bn+c$ . Put row numbers in the input column and bead counts in the output column, then evaluate the rule at the requested row. The tempting shortcut is to assume a constant increase between rows, but a quadratic's increases can change.

Hints

Hint 1
A function value like $B(4)=49$ pairs an input, row $4$ , with an output, $49$ beads. In a Desmos table, which column should hold the row numbers, and which should hold the bead counts?
Hint 2
A quadratic has the form $an^2+bn+c$ , with three coefficients to find. Fit all three given pairs together rather than using two pairs to assume a constant increase.
Hint 3
After the fit, define the function using Desmos's stored coefficients. Evaluating $B(3)$ means putting row number $3$ into that function, not reading the third entry in the table.

Step-by-step

Fit the quadratic in Desmos
Step 1
Pair each row with its bead count
$B(1)=10$ means row $1$ has $10$ beads, so put row numbers in Desmos's $x_1$ column and counts in its $y_1$ column. Enter the three pairs as shown. Keep each input and output on the same table row; Desmos will use those pairs to fit the model.
Step 2
Find the three coefficients
Type $y_1\sim ax_1^2+bx_1+c$ . The $\sim$ tells Desmos to fit the quadratic to the table. Three distinct inputs determine one quadratic, so Desmos reports $a=2$ , $b=3$ , and $c=5$ under PARAMETERS.
Step 3
Define the bead-count function
Type $B(n)=an^2+bn+c$ , using the stored coefficients Desmos found. Here that defines $B(n)=2n^2+3n+5$ , where $n$ is the row number and $B(n)$ is its bead count. Using the stored letters also avoids rounding errors in problems with decimal coefficients.
Step 4
Find the count in row 3
Type $B(3)$ to put row $3$ into the fitted function. Desmos prints $32$ . So row $3$ has $32$ beads. Choice B.

Question 12·SAT Desmos Regression Practice·Advanced Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this