Question

describe how the graph of the function $y = \frac{4}{7} (x - 1)^2 + 3$ is related to the graph of the parent function.
vertical translation: select choice
horizontal translation: select choice
dilation: select choice
reflection: select choice

Explanation:

Step1: Identify parent function

The parent function is $y=x^2$.

Step2: Analyze vertical translation

Look at the constant term $+3$. For $y=a(x-h)^2+k$, $k>0$ means up shift.
Vertical translation: 3 units up

Step3: Analyze horizontal translation

Look at $(x-1)$. For $y=a(x-h)^2+k$, $h>0$ means right shift.
Horizontal translation: 1 unit right

Step4: Analyze dilation

Look at the coefficient $\frac{4}{7}$. Since $0<|a|<1$, it's a vertical compression.
Dilation: Vertical compression by factor $\frac{4}{7}$

Step5: Analyze reflection

The coefficient $\frac{4}{7}$ is positive, so no reflection over x-axis.
Reflection: No reflection

Answer:

vertical translation: 3 units up
horizontal translation: 1 unit right
dilation: Vertical compression by a factor of $\frac{4}{7}$
reflection: No reflection across the x-axis