Question

Given the equation \(y=3 x^2+4\), what is the function of the coefficient of 3 ? A) It moves the graph of \(y=3 x^2+4\) three units higher than the graph of \(y=x^2+4\). B) It moves the graph of \(y=3 x^2+4\) three units lower than the graph of \(y\) \(=x^2+4\) C) It makes the graph of \(y=3 x^2+4\) wider than the graph of \(y=x^2+4\). D) It makes the graph of \(y=3 x^2+4\) narrower than the graph of \(y=x^2+\) 4.

Short answer

D) It makes the graph of \(y=3x^2+4\) narrower than the graph of \(y=x^2+4\).

Step-by-step solution

Understand the effect of the coefficient of 3

The coefficient of \(3\) affects the parabola's shape by multiplying the quadratic term. This results in the change of the graph's width, either widening or narrowing the parabola. In this case, since \(3 > 1\), it will make the graph narrower.

Compare the answer choices

Now that we have determined that the coefficient of 3 makes the graph narrower, let's compare this to the answer choices given. A) It moves the graph three units higher - This is incorrect, as the coefficient of 3 does not affect the vertical shift of the graph. B) It moves the graph three units lower - This is also incorrect for the same reason as A. C) It makes the graph wider - This is incorrect, as we have determined that the graph becomes narrower. D) It makes the graph narrower - This matches with our analysis, and thus, is the correct answer. So, the correct answer is: D) It makes the graph of \(y=3x^2+4\) narrower than the graph of \(y=x^2+4\).