Question

the\ function\ h\left(x\right)=\frac{1}{2}\left(x+3\right)^{2}+2.How\ is\ the\ graph\ of\ h\left(x\right)\ t\ r\ a\ nslated\ \ from\ the\ parent\ graph\ of\ a\ quadratic\ function,\ f\left(x\right)=x^{2}

Answer 1

The function h(x) is vertically stretched by a factor of $\frac{1}{2}$ and is shifted 3 units to the left and shifted 2 units up.

Explanation:

The parent function is $f(x)=x^{2}$

The translated function is $h(x)=\frac{1}{2}(x+3)^{2}+2$

By using the function transformation rules, the translated function h(x) is stretched vertically by a factor of $\frac{1}{2}$

Also, from the function transformation rules, we know that,  $f(x+b)$ shifts the function b units to the left.

Thus, the translated function h(x) is shifted 3 units to the left.

By using the function transformation rules, we know that,  $f(x)+b$ shifts the function b units upward.

Thus, the translated function h(x) is shifted 2 units upwards.

Thus, the translated function h(x) is vertically stretched by a factor of $\frac{1}{2}$ and is shifted 3 units to the left and shifted 2 units up.