Question

the function $g(x) = 10x^2 - 100x + 213$ written in vertex form is $g(x) = 10(x - 5)^2 - 37$. which statements are true about $g(x)$? select three options.

• the axis of symmetry is the line $x = -5$.
• the vertex of the graph is $(5, -37)$.
• the parabola has a minimum.
• the parabola opens up.
• the value of $a$, when the equation is written in vertex form, is negative.

Explanation:

Step1: Recall vertex form rules

For $g(x)=a(x-h)^2+k$, vertex is $(h,k)$, axis is $x=h$.

Step2: Identify $a, h, k$

From $g(x)=10(x-5)^2-37$, $a=10$, $h=5$, $k=-37$.

Step3: Verify axis of symmetry

Axis is $x=5$, so $x=-5$ is false.

Step4: Verify vertex

Vertex is $(h,k)=(5,-37)$, so this is true.

Step5: Check parabola direction

$a=10>0$, so parabola opens up, has a minimum. These are true.

Step6: Check sign of $a$

$a=10$ is positive, so "negative $a$" is false.

Answer:

• The vertex of the graph is (5, -37).
• The parabola has a minimum.
• The parabola opens up.