The graph of the function $\newline$ $y=x^{2}$ is shown. How will the graph change if the equation is changed to $\newline$ $y=\left(\frac{1}{4}\right)x^{2}$ ? $\newline$ The parabola will become narrower. $\newline$ The parabola will move up $\frac{1}{4}$ unit. $\newline$ The parabola will become wider. $\newline$ The parabola will move down $\newline$ $\frac{1}{4}$ unit.

Full solution

Q. The graph of the function

\newline

y=x^{2}

is shown. How will the graph change if the equation is changed to

\newline

y=\left(\frac{1}{4}\right)x^{2}

\newline

The parabola will become narrower.

\newline

The parabola will move up

\frac{1}{4}

unit.

\newline

The parabola will become wider.

\newline

The parabola will move down

\newline

\frac{1}{4}

unit.

Compare Equations: Compare the original equation $y=x^2$ with the new equation $y=\frac{1}{4}x^2$ . The coefficient of $x^2$ in the original equation is $1$ , which means the parabola is in its standard width. The coefficient of $x^2$ in the new equation is $\frac{1}{4}$ , which is less than $1$ . A smaller coefficient in front of $x^2$ indicates that the parabola will open more widely.
Coefficient Analysis: Determine the effect of changing the coefficient on the width of the parabola. When the coefficient of $x^2$ is reduced from $1$ to $\frac{1}{4}$ , the parabola becomes wider because the same $y$ -value is reached for a larger range of $x$ -values. This means that the graph of $y=\left(\frac{1}{4}\right)x^2$ will be wider than the graph of $y=x^2$ .
Vertical Shift Check: Check if there is any vertical shift in the graph. $\newline$ Since there is no constant term added or subtracted from the equation $y=\frac{1}{4}x^2$ , there is no vertical shift up or down. $\newline$ The vertex of the parabola remains at the origin $(0,0)$ .
Eliminate Incorrect Options: Eliminate incorrect options based on the analysis. $\newline$ The parabola will not become narrower, so we can eliminate that option. $\newline$ The parabola will not move up or down, so we can eliminate the options that suggest a vertical shift. $\newline$ The correct answer is that the parabola will become wider.