Question

homework assignment 3.2: quadratic functions
score: 2/10 answered: 3/10
question 4
consider the quadratic function $f(x) = x^2 - 7x - 8$.
its vertex is ( , ).
its largest $x$-intercept is $x = $
its $y$-intercept is $(0, )$.
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Explanation:

Step1: Find the x-coordinate of the vertex

For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = -7 \), so \( x = -\frac{-7}{2\times1} = \frac{7}{2} = 3.5 \).

Step2: Find the y-coordinate of the vertex

Substitute \( x = 3.5 \) into \( f(x) \): \( f(3.5) = (3.5)^2 - 7\times3.5 - 8 = 12.25 - 24.5 - 8 = -20.25 \). So the vertex is \( (3.5, -20.25) \).

Step3: Find the x-intercepts

Set \( f(x) = 0 \): \( x^2 - 7x - 8 = 0 \). Factor: \( (x - 8)(x + 1) = 0 \). So \( x = 8 \) or \( x = -1 \). The largest x-intercept is \( x = 8 \).

Step4: Find the y-intercept

Set \( x = 0 \) in \( f(x) \): \( f(0) = 0^2 - 7\times0 - 8 = -8 \). So the y-intercept is \( (0, -8) \).

Answer:

Vertex: \( (3.5, -20.25) \)
Largest x-intercept: \( x = 8 \)
Y-intercept: \( (0, -8) \)