Question

gerald graphs the function $f(x) = (x - 3)^2 - 1$. which statements are true about the graph? choose three correct answers.
the domain is ${x | x \geq 3}$.
the function decreases over the interval $(-\infty, 3)$
the vertex is $(3, -$
the axis of symmetry is $x = -1$.
the range is ${y | y \geq -1}$.

Explanation:

Step1: Identify vertex form properties

The function is in vertex form $f(x) = (x-h)^2 + k$, where $(h,k)$ is the vertex, axis of symmetry is $x=h$. For $f(x)=(x-3)^2-1$, $h=3$, $k=-1$. So vertex is $(3,-1)$, axis of symmetry $x=3$.

Step2: Analyze domain

Quadratic functions have no restrictions on $x$, so domain is all real numbers, not $\{x | x \geq 3\}$.

Step3: Analyze increasing/decreasing intervals

Since the coefficient of $(x-3)^2$ is positive, the parabola opens upward. It decreases on $(-\infty, h)=(-\infty, 3)$ and increases on $(3, \infty)$.

Step4: Analyze range

The minimum value of the function is $k=-1$, so range is $\{y | y \geq -1\}$.

Step5: Verify axis of symmetry

Axis of symmetry is $x=3$, not $x=-1$.

Answer:

1. The function decreases over the interval $(-\infty, 3)$
2. The vertex is $(3, -1)$
3. The range is $\{y \mid y \geq -1\}$