Question

analyzing the graph of a quadratic function the function ( f(x) = -0.3(x - 5)^2 + 5 ) is graphed. what are some of its key features? check all that apply.

• the axis of symmetry is ( x = 5 ).
• the domain is ( {x | x \text{ is a real number}} ).
• the function is increasing over ( (-infty, 5) ).
• the minimum is ( (5, 5) ).
• the range is ( {y | y geq 5} ).

Explanation:

Step1: Identify quadratic form

The function is in vertex form $f(x)=a(x-h)^2+k$, where $a=-0.3$, $h=5$, $k=5$.

Step2: Check axis of symmetry

Axis of symmetry is $x=h$, so $x=5$.

Step3: Check domain

Quadratic functions have all real $x$ as domain.

Step4: Check increasing interval

Since $a<0$, the parabola opens downward. It increases left of the vertex, so over $(-\infty, 5)$.

Step5: Check minimum/maximum

Downward opening parabola has a maximum at $(h,k)=(5,5)$, no minimum.

Step6: Check range

For downward opening parabola, $y\leq k$, so $y\leq5$.

Answer:

• The axis of symmetry is $x = 5$.
• The domain is $\{x| x \text{ is a real number}\}$.
• The function is increasing over $(-\infty, 5)$.