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:

1. For the axis of symmetry: The vertex form of a parabola is \(y = a(x - h)^2 + k\), where the axis of symmetry is \(x = h\). Here, \(h = 5\), so the axis of symmetry is \(x = 5\), not \(x=-5\). So the first statement is false.
2. For the vertex: In the vertex form \(y = a(x - h)^2 + k\), the vertex is \((h,k)\). Here, \(h = 5\) and \(k=-37\), so the vertex is \((5, - 37)\). So the second statement is true.
3. For the direction of the parabola and minimum/maximum: The coefficient \(a\) in the vertex form \(y=a(x - h)^2 + k\) determines the direction. If \(a>0\), the parabola opens up and has a minimum (since it opens upwards, the vertex is the lowest point). Here, \(a = 10>0\), so the parabola opens up and has a minimum. So the third and fourth statements are true.
4. For the value of \(a\): In the vertex form \(g(x)=10(x - 5)^2-37\), \(a = 10\) which is positive, not negative. So the fifth statement is false.

Answer:

B. The vertex of the graph is \((5, -37)\)
C. The parabola has a minimum.
D. The parabola opens up.