Question

Which of the following parabolas opens upward and appears narrower than y = −3x^2 + 2x − 1?

Answer 1

To find a parabola that opens upward and is narrower than the given equation, look for a positive coefficient of the x² term that is greater than 3 in absolute value.

Step-by-step explanation:

The student's question asks which of the given parabolas opens upward and is narrower than y = −3x² + 2x − 1. To determine this, we must compare the coefficients of the x² term in the parabolic equations. A parabola will open upwards if the coefficient of the x² term is positive, and the parabola will appear narrower if this coefficient has a larger absolute value compared to another parabola with an upward opening.

In the equation y = −3x² + 2x − 1, the coefficient of x² is −3, which means this parabola opens downward. For a parabola to open upwards and be narrower than this, it would have to have a positive coefficient that is larger in magnitude than 3.

Answer 2

The key to this question: Do you see the number that comes right before the
`x^(2)` term?

If that number has a negative sign before it, it is negative and the graph will point downwards. However, if it is a positive number, the graph will point downwards.

You can now eliminate options 2 and 4, because they are negative and you want a graph that is positive.

Now look at how large that integer is. Ignore the negative sign completely if there is one. The larger it is, the narrower your graph will be. Imagine the slope of a normal line. If the slope is a larger number, it will be more steep and therefore more narrow.

An equation is quite literally altering the x value to get a y value. If x = y, for every x value you have, y will be exactly the same. However, if you multiply x by a number, like
`\frac{1}4}` or 5, the y value will then become smaller or bigger. When this happens, the graph becomes wider or narrower. When that integer is larger, the graph will become narrower.

Now take your equation y = −3x^2 + 2x − 1.

Because you want a graph that is narrow, find an integer that is larger than 3.

The only option with an integer that is larger than 3 and is positive is Option 1.

The answer is Option 1, y = 4x2 − 2x − 1