5. which graph matches the function ( h(x)=-\frac{1}{3} x^{2} ) ?\n6. what are the transformations of the…

Question 6

Brief Explanations:

For a quadratic function (y = a(x - h)^2 + k) (in this case (h = 0,k = 0)), the coefficient (a) determines transformations. If (a=- \frac{1}{3}), the negative sign indicates a vertical reflection (flips the parabola over the (x) - axis). The value (\vert a\vert=\frac{1}{3}<1) indicates a vertical shrink. A horizontal reflection would be of the form (y = f(-x)) (e.g., (y=(-x)^2=x^2) for the parent function, which is not the case here). A vertical stretch would be when (\vert a\vert> 1).

Answer:

B. vertical reflection, vertical shrink of (\frac{1}{3})

Question 7

Explanation:

Step1: Use the factored form of a quadratic

If the (x) - intercepts are (x = p) and (x = q), the quadratic function in factored form is (y=a(x - p)(x - q)). Here (p=-5) and (q = 8), so (y=a(x + 5)(x - 8)). Expanding ((x + 5)(x - 8)=x^{2}-8x+5x - 40=x^{2}-3x - 40) (when (a = 1))

Answer:

A. (y=x^{2}-3x - 40)

Question 8

Explanation:

Step1: Set the function equal to zero

Set (f(x)=3x^{2}-6x - 45 = 0).

Step2: Factor out the greatest common factor

Factor out (3): (3(x^{2}-2x - 15)=0). Then, since (3\neq0), we focus on (x^{2}-2x - 15=0).

Step3: Factor the quadratic

Factor (x^{2}-2x - 15): we need two numbers that multiply to (-15) and add to (-2). The numbers are (-5) and (3), so (x^{2}-2x - 15=(x - 5)(x+3)=0).

Step4: Solve for (x)

Using the zero - product property (x - 5=0) gives (x = 5) and (x+3=0) gives (x=-3)

Answer:

D. (x=-3,x = 5)