Question

what are the key points on the graph of $y = x^2 - 16x + 28$? name the vertex, x-intercepts, and y-intercept. (2 points)

vertex: (□)

(smaller) x-intercept: (□)

(larger) x-intercept: (□)

y-intercept: (□)

Explanation:

Step1: Find the vertex (h,k)

For a quadratic function \( y = ax^2 + bx + c \), the x-coordinate of the vertex \( h = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = -16 \), so \( h = -\frac{-16}{2\times1} = 8 \). Substitute \( x = 8 \) into the function: \( y = 8^2 - 16\times8 + 28 = 64 - 128 + 28 = -36 \). So vertex is \( (8, -36) \).

Step2: Find x-intercepts (set y=0)

Solve \( x^2 - 16x + 28 = 0 \). Factor: \( (x - 2)(x - 14) = 0 \). So \( x - 2 = 0 \) or \( x - 14 = 0 \), giving \( x = 2 \) and \( x = 14 \). Smaller x-intercept is \( 2 \), larger is \( 14 \).

Step3: Find y-intercept (set x=0)

Substitute \( x = 0 \) into the function: \( y = 0^2 - 16\times0 + 28 = 28 \). So y-intercept is \( (0, 28) \).

Answer:

vertex: \((8, -36)\)
(smaller) x-intercept: \((2, 0)\)
(larger) x-intercept: \((14, 0)\)
y-intercept: \((0, 28)\)