which statements are true about the graph of the function f(x) = x² - 8x + 5? select three options. the…

Explanation:

Step1: Convert to vertex - form

For a quadratic function (y = ax^{2}+bx + c), the vertex - form is (y=a(x - h)^{2}+k), where (h=-\frac{b}{2a}) and (k = y(h)). Given (f(x)=x^{2}-8x + 5), (a = 1), (b=-8), (c = 5). First, find (h): (h=-\frac{-8}{2\times1}=4). Then substitute (x = 4) into (f(x)): (f(4)=4^{2}-8\times4 + 5=16-32 + 5=-11). So (f(x)=(x - 4)^{2}-11), the first option is true.

Step2: Find the vertex

The vertex of the quadratic function in vertex - form (y=a(x - h)^{2}+k) is ((h,k)). For (f(x)=(x - 4)^{2}-11), the vertex is ((4,-11)), not ((-8,5)), so the second option is false.

Step3: Find the axis of symmetry

The axis of symmetry of a quadratic function (y = ax^{2}+bx + c) is (x=-\frac{b}{2a}). Since (a = 1) and (b=-8), (x=-\frac{-8}{2\times1}=4), not (x = 5), so the third option is false.

Step4: Find the y - intercept

To find the (y) - intercept, set (x = 0) in (f(x)). Then (f(0)=0^{2}-8\times0 + 5=5). So the (y) - intercept is ((0,5)), the fourth option is true.

Step5: Find the x - intercepts

Set (f(x)=0), so (x^{2}-8x + 5=0). The discriminant (\Delta=b^{2}-4ac=(-8)^{2}-4\times1\times5=64 - 20 = 44>0). Since (\Delta>0), the function crosses the (x) - axis twice, the fifth option is true.

Answer:

The function in vertex form is (f(x)=(x - 4)^{2}-11); The (y) - intercept of the function is ((0,5)); The function crosses the (x) - axis twice.