How can the $y$-intercept be negative if the constant in the equation is positive?

Question

Line $F$ can be described by the function $f(x)=5x$. Line $G$ is parallel to Line $F$ such that the shortest distance between Line $G$ and Line $F$ is $c$, and the $y$-intercept of Line $G$ is negative. Which of the following is a possible equation for line $G$?

• A: $g(x)=x-5$
• ✔: $g(x)+5\sqrt2=5x$
• C: $g(x)=x-5\sqrt2$
• ❌: $g(x)-5=5x$

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Isn’t the answer choice D if the y-intercept is negative?

Answer 1

There are a couple ways to understand the $y$-intercepts here.

Substitute $0$ for $x$

The first way is to recall the meaning of $y$-intercept. It's the height (or the point, depending on the source) where the graph touches ("intercepts") the $y$-axis. And the $y$-axis is all the points where $x=0$.

So if we substitute $0$ in for $x$ in an equation, we find out what's true for the graph on the $y$-axis. Doing this for all four choices, we have:

• A: $g(0)=0-5$
• B: $g(0)+5\sqrt2=5*0$
• C: $g(0)=0-5\sqrt2$
• D: $g(0)-5=5*0$

Since we want the height, we want the value of $g$, $g(0)$. So let's find out what $g(0)$ is in each case (some would say "let's solve for $g(0)$").

• A: $g(0)=0-5=-5$
• B: $g(0)=5*0-5\sqrt2=-5\sqrt2$
• C: $g(0)=0-5\sqrt2=-5\sqrt2$
• D: $g(0)=5*0+5=5$

From this, we see that choice D is the only one with a positive $y$-intercept, so it can't be the right answer.

Use $y=mx+b$ form

The other way, which only works when you have the entire equation of a line (but is fine for this problem since you also care about the slope), is to manipulate the equation into a standard form "$y=mx+b$". Recall that $m$ is the slope and $b$ is the $y$-intercept when you have an equation of a line like that.

Since we have a function like $g(x)$, the graph of the function would have the outputs as $y$ coordinates, so we can start by writing $y$ in place of $g(x)$ in each choice:

• A: $y=x-5$
• B: $y+5\sqrt2=5x$
• C: $y=x-5\sqrt2$
• D: $y-5=5x$

To make these look like $y=mx+b$, we need to put $y$ on one side (to solve for $y$). We might also want to write any subtraction as adding a negative so we get the $+$ in the $+b$. If we want the slope, we could also write an isolated $x$ as $1x$ so we can read off the $m$ in that case.

• A: $y=x-5$ or $y=1x+(-5)$
• B: $y=5x-5\sqrt2$ or $y=5x+(-5\sqrt2)$
• C: $y=x-5\sqrt2$ or $y=1x+(-5\sqrt2)$
• D: $y=5x+5$

So we can read off the intercepts and see that choice D is the only one with a positive intercept.