## What’s up (more), the parabola or the line?

## Question

The graph to the left shows the function *f*(*x*), a parabola – that is, a quadratic function. Not shown on the graph is function in which *c* is a negative constant.

One statement below is true. Which one?

## Solution

What’s going on? is the equation of a straight line. The line will cross the parabola in a couple of places and those intersections will determine which function is higher in various places. But all you know about function is that and that *c* is negative.

Think of a line’s basic equation: where *b* is the line’s *y*-intercept – that is, *y*’s value where . In this case, because nothing, the *y*-intercept is 0. That means **the line goes through the origin.**

Also, you’re told that *c* is negative. *c* is in the role of *m* in the equation that is, it’s the slope. The negative slope means **this line goes down as it goes to the right**.

How steep is the line? You are given no information about that. Take a guess.

The brown line in the graph to the right shows what function *g* may look like. It may be steeper or less steep than shown, but it will cross the *x*-axis at the origin, and it will go down as it goes to the right, as shown.

Let’s go through all the possible answers to find the one true statement.

- . At , as you can see from the graph, the parabola is above the
*x*-axis, which means*f*(0) is a positive value. The line goes right through the origin – so . Thus which means it cannot be true that . - . At , function
*f*crosses the*x*-axis. So . At , function*g*crosses the*x*-axis. So . That means*f*(*a*) is equal to*g*(0), so it’s not possible that*f*(*a*) is less than*g*(0). - . What’s happening at ? Function
*f*crosses the*x*-axis there, so ; and function*g*has a positive value. That means*g*is more than*f*, so it must be that . Looks like that’s the answer, but let’s still check the others. - . We have established that . And since
*g*(*x*) goes through the origin and slopes down,*g*must be less than zero at any*x*-value that is greater than 0 – like, for example,*b*. That means*g*is less than*f*, so it is not possible that*f*is less than*g*. - . As you can see from the graph, . And, as we said in response to answer #4,
*g*(*b*) is less than 0. So*f*(*b*) must be more than*g*(*b*) – and it’s not possible that .

So answer 3, and only answer 3, works.

I found this question intimidating, because it seems so difficult to know anything about function *g*. But once you look closely, it’s not so bad. The key is to notice that *g* takes on positive values to the left of and negative values to the right of .