Question

shifting the cube root function quick check
what is the effect on the y-intercept in the graph of $f(x) = \sqrt3{x}$ when $f(x)$ is replaced by $f(x) + 5$? (1 point)
\bigcirc the y-intercept shifts up by a distance of 5.
\bigcirc the y-intercept shifts to the right by a distance of 5.
\bigcirc the y-intercept shifts down by a distance of 5.
\bigcirc the y-intercept shifts to the left by a distance of 5.

Explanation:

Step1: Find original y - intercept

For the function \(f(x)=\sqrt[3]{x}\), to find the y - intercept, we set \(x = 0\). Then \(f(0)=\sqrt[3]{0}=0\). So the y - intercept of the original function is at the point \((0,0)\).

Step2: Find new y - intercept

When we replace \(f(x)\) with \(f(x)+5\), the new function is \(g(x)=f(x)+5=\sqrt[3]{x}+5\). To find the y - intercept of the new function, we set \(x = 0\). Then \(g(0)=\sqrt[3]{0}+5=0 + 5=5\). So the y - intercept of the new function is at the point \((0,5)\).

Step3: Analyze the shift

The original y - intercept was at \((0,0)\) and the new y - intercept is at \((0,5)\). In the coordinate plane, a change from \(y = 0\) to \(y=5\) for the same \(x = 0\) means the point has moved up by 5 units.

Answer:

The y - intercept shifts up by a distance of 5.