Question 29·SAT Desmos Regression Practice·Algebra

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The function $h$ is linear. If $h(-3)=12$ and $2h(5)-h(-1)=30$ , which choice is the value of $h(0)$ ?

A linear function has a slope and a value at input $0$ . When you're given two conditions involving its outputs, write $h(x)=mx+b$ and use a matched-list regression in Desmos to find both unknowns. The key trap is treating an output at a nonzero input as the value at $0$ .

Hints

Hint 1
In the linear rule $h(x)=mx+b$ , $m$ is the slope and $b$ is the output at input $0$ . The question asks for $h(0)$ , so which letter do you need to find?
Hint 2
Each expression such as $h(-3)$ means put that input into the same rule. Replace the function notation in both conditions, keeping the $2$ in front of the entire output $h(5)$ .
Hint 3
A matched-list regression pairs each expression with its given value. Put both conditions in one regression so Desmos finds one shared slope and value at $0$ .

Step-by-step

Fit the linear rule in Desmos
Step 1
Identify the value you need
A linear function can be written $h(x)=mx+b$ , where $m$ is its slope and $b$ is its output at input $0$ . So $h(0)=m(0)+b=b$ . The output at input $0$ is $b$ , not an output at another input.
Step 2
Turn the first output into an equation
$h(-3)=12$ means putting in $-3$ gives $12$ , so substitute into the rule: $-3m+b=12$ . Don't set $b=12$ : the given input is $-3$ , not $0$ .
Step 3
Turn the combined outputs into an equation
Here $h(5)=5m+b$ and $h(-1)=-m+b$ . The $2$ multiplies all of $h(5)$ , while all of $h(-1)$ is subtracted, so: $2(5m+b)-(-m+b)=30$ . Keep the parentheses around $-m+b$ so the subtraction applies to both terms.
Step 4
Fit both conditions and read the value at zero
Type $[-3m+b,2(5m+b)-(-m+b)]\sim[12,30]$ . A regression uses $\sim$ to match the entries in order with one shared $m$ and $b$ . Desmos shows $m\approx1.285714$ and $b\approx15.857143$ under PARAMETERS. Since $h(0)=b$ and $\frac{111}{7}\approx15.857143$ , the value of $h(0)$ is $\frac{111}{7}$ . Choice B.

Question 29·SAT Desmos Regression Practice·Algebra

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