Question 377·Easy·Algebra

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The function $g$ is linear with slope $-2$ and satisfies $g(2)=-1$ . Which of the following defines $g$ ?

A linear function has a constant slope, so a slope and one input-output pair determine its rule. Write the rule as $g(x)=mx+b$ , then use the given function value to find $b$ with a Desmos regression. The tempting slip is to treat the output at a nonzero input as $b$ , even though $b$ is the output at input $0$ .

Hints

Hint 1
In slope-intercept form, $g(x)=mx+b$ , $m$ is the slope and $b$ is the output when the input is $0$ . The given slope tells you which number to put in for $m$ ?
Hint 2
$g(2)=-1$ says input $2$ gives output $-1$ . Put both numbers into your rule. The output at $2$ is not automatically the intercept, which is the output at $0$ .

Step-by-step

Use the slope and function value
Step 1
Put the slope into the rule
Use slope-intercept form, $g(x)=mx+b$ . Here $m$ is the slope, and $b$ is the output when $x=0$ . The given slope $-2$ makes the rule $g(x)=-2x+b$ .
Step 2
Turn the function value into an equation
$g(2)=-1$ means the input $2$ gives output $-1$ . Put those numbers into the rule: $-1=-2(2)+b$ . The output at $2$ is not $b$ ; $b$ is the output at $0$ .
Step 3
Find the missing intercept
Use a regression: type $-1\sim-2(2)+b$ in Desmos. The tilde, $\sim$ , tells Desmos to find the unknown $b$ . Under PARAMETERS, it shows $b=3$ , so the output at $x=0$ is $3$ .
Step 4
Match the complete rule
Replace $b$ with $3$ : $g(x)=-2x+3$ . This rule has the required slope and gives the required output at input $2$ , so it defines $g$ . Choice D.

Question 377·Easy·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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