Question

determining characteristics of a graph from an equation
which are characteristics of the graph of the function ( f(x) = (x + 1)^2 + 2 )? check all that apply

• the domain is all real numbers.
• the range is all real numbers greater than or equal to 1
• the ( y )-intercept is 3.
• the graph of the function is 1 unit up and 2 units to the left from the graph of ( y = x^2 )
• the graph has two ( x )-intercepts.

Explanation:

Step1: Analyze the domain

For any quadratic function $f(x)=(x+1)^2+2$, there are no restrictions on $x$ (no division by zero, no square roots of negative numbers for real $x$). So the domain is all real numbers.

Step2: Calculate the range

The term $(x+1)^2$ is always non-negative for real $x$, so $(x+1)^2\geq0$. Add 2 to both sides: $(x+1)^2+2\geq2$. Thus the range is all real numbers $\geq2$.

Step3: Find the y-intercept

Set $x=0$: $f(0)=(0+1)^2+2=1+2=3$. So the y-intercept is 3.

Step4: Analyze the transformation

The parent function is $y=x^2$. The transformed function is $f(x)=(x-(-1))^2+2$. This represents a shift 1 unit left (from $x-(-1)$) and 2 units up (from $+2$), not 1 up and 2 left.

Step5: Check x-intercepts

Set $f(x)=0$: $(x+1)^2+2=0 \implies (x+1)^2=-2$. Since a square of a real number cannot be negative, there are no real x-intercepts.

Answer:

• The domain is all real numbers.
• The y-intercept is 3.