Question

1. what is the correct quadratic function for this parabola? (image of parabola)

a. $f(x) = (x - 2)(x - 3)$
b. $f(x) = (x + 2)(x - 3)$
c. $f(x) = (x - 2)(x + 3)$
d. $f(x) = (x + 2)(x + 3)$

1. which set of data is correct for the quadratic relation $f(x) = -2(x - 12)^2 + 15$?

	direction parabola opens	vertex	axis of symmetry
a.	downward	$(15, -12)$	$x = 15$
b.	downward	$(12, 15)$	$x = 12$
c.	upward	$(-12, 15)$	$x = -12$
d.	upward	$(15, 12)$	$x = 15$

a. set d.
b. set b.
c. set a.
d. set c.

1. which function has a minimum value?

a. $f(x) = (x - 5)^2 + 15$
b. $f(x) = -(x + 1)^2 - 5$
c. $f(x) = -(x - 15)^2 + 5$
d. $f(x) = -(x - 5)^2 + 10$

Explanation:

Question 9

Step1: Identify x-intercepts

From the graph, the parabola crosses the x - axis at \(x=-3\) and \(x = 2\). So the roots of the quadratic function are \(x=-3\) (which means \(x + 3=0\)) and \(x = 2\) (which means \(x-2 = 0\)).

Step2: Write the factored form

The factored form of a quadratic function with roots \(r_1\) and \(r_2\) is \(f(x)=(x - r_1)(x - r_2)\). Substituting \(r_1=-3\) and \(r_2 = 2\), we get \(f(x)=(x - 2)(x+3)\).

Step1: Analyze the vertex form \(f(x)=a(x - h)^2+k\)

For the quadratic function \(f(x)=-2(x - 12)^2+15\), the vertex form is \(y=a(x - h)^2+k\), where \((h,k)\) is the vertex and \(a\) determines the direction of opening.

• If \(a>0\), the parabola opens upward; if \(a < 0\), it opens downward. Here \(a=-2<0\), so the parabola opens downward.
• The vertex \((h,k)=(12,15)\).
• The axis of symmetry is \(x = h=x = 12\).

Step2: Match with the options

Looking at the table, Set B has direction downward, vertex \((12,15)\) and axis of symmetry \(x = 12\).

Step1: Recall the vertex form of a quadratic function

For a quadratic function in the form \(f(x)=a(x - h)^2+k\):

• If \(a>0\), the parabola opens upward, and the function has a minimum value at \(x = h\) (the vertex is the minimum point).
• If \(a<0\), the parabola opens downward, and the function has a maximum value at \(x = h\) (the vertex is the maximum point).

Step2: Analyze each option

• Option a: \(f(x)=(x - 5)^2+15\), here \(a = 1>0\), so it opens upward and has a minimum value.
• Option b: \(f(x)=-(x + 1)^2-5\), \(a=-1<0\), opens downward (has a maximum).
• Option c: \(f(x)=-(x - 15)^2+5\), \(a=-1<0\), opens downward (has a maximum).
• Option d: \(f(x)=-(x - 5)^2+10\), \(a=-1<0\), opens downward (has a maximum).

Answer:

c. \(f(x)=(x - 2)(x + 3)\)

Question 10