Question

(b) the graph of ( y = x^2 ) is shown. translate it to get the graph of ( y = (x + 4)^2 ).

Explanation:

Step1: Identify translation rule

For a function $y = f(x + h)$, if $h>0$, the graph of $y=f(x)$ shifts left by $h$ units. Here, $f(x)=x^2$, $h=4$.

Step2: Apply translation to key points

Take vertex of $y=x^2$: $(0,0)$. Shift left 4 units: $(0-4, 0)=(-4,0)$.
Take other points e.g., $(1,1)$: shift to $(1-4,1)=(-3,1)$; $(-1,1)$: shift to $(-1-4,1)=(-5,1)$; $(2,4)$: shift to $(2-4,4)=(-2,4)$; $(-2,4)$: shift to $(-2-4,4)=(-6,4)$.

Step3: Plot translated points

Plot $(-4,0), (-3,1), (-5,1), (-2,4), (-6,4)$ and draw the parabola through them, matching the shape of $y=x^2$.

Answer:

The graph of $y=(x+4)^2$ is the graph of $y=x^2$ translated 4 units to the left, with vertex at $(-4,0)$ and the same parabolic shape as the original function. When plotted on the grid, all points of $y=x^2$ are shifted horizontally left by 4 units.