Question 76·Easy·Linear Functions

Look at the coefficient of $x$ in each answer option. Cross out any choices that do not have a coefficient of $-2$.

For the options that are left, plug in $x=2$ and see what $g(2)$ equals. Which option gives an output of $-1$?

Type each option into Desmos exactly as written: y = -2x - 3, y = 2x + 3, y = 2x - 3, and y = -2x + 3. Each will appear as a different line on the graph.

Look at the lines and note which ones slope downward from left to right (negative slope) and which slope upward (positive slope). Only the lines with negative slope could match a slope of $-2$.

In Desmos, click on or add the point $(2,-1)$ (you can type (2,-1) as a point). See which of the lines passes exactly through this point. The equation of that line is the function that correctly defines $g$.

A linear function in slope-intercept form looks like $y=mx+b$, where $m$ is the slope.

• Here, the slope is $-2$, so we need $m=-2$.
• That means we are only interested in answer choices where the coefficient of $x$ is $-2$.

From the options:

• $g(x)=-2x-3$ has slope $-2$.
• $g(x)=-2x+3$ has slope $-2$.
• The other two choices have slope $2$, so they cannot be correct.

We know the function passes through the point $(2,-1)$, which means when $x=2$, the output (or $y$-value) must be $-1$.

Take each remaining option and substitute $x=2$:

• For $g(x)=-2x-3$, compute $g(2)$.
• For $g(x)=-2x+3$, compute $g(2)$.

Only one of these will give $-1$.

Now calculate each value:

• For $g(x)=-2x-3$: $g(2)=-2(2)-3=-4-3=-7$, which is not $-1$.
• For $g(x)=-2x+3$: $g(2)=-2(2)+3=-4+3=-1$, which matches $g(2)=-1$.

So the function that fits both the slope and the point is $g(x)=-2x+3$.