Question

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Explanation:

Step1: Find the vertex (a)

For a quadratic function \( f(x) = ax^2 + bx + c \), the x - coordinate of the vertex is \( x = -\frac{b}{2a} \). Here, \( a=-1 \), \( b = - 4 \), so \( x=-\frac{-4}{2\times(-1)}=-2 \). Then substitute \( x = - 2 \) into \( f(x) \): \( f(-2)=-(-2)^2-4\times(-2)+21=-4 + 8+21 = 25 \). So the vertex is \( (-2,25) \).

Step2: Find the y - intercept (b)

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0)=-0^2-4\times0 + 21=21 \). So the y - intercept is \( (0,21) \).

Step3: Determine maximum/minimum (c)

Since \( a=-1<0 \), the parabola opens downwards, so the function has a maximum value. The maximum value is the y - coordinate of the vertex, which is 25. So the first blank is "maximum" and the second is 25.

Step4: Find x - intercepts (d)

Set \( f(x)=0 \), so \( -x^2-4x + 21 = 0 \). Multiply both sides by - 1: \( x^2 + 4x-21=0 \). Factor: \( (x + 7)(x - 3)=0 \). So \( x=-7 \) or \( x = 3 \). The x - intercepts are \( (-7,0) \) and \( (3,0) \).

Step5: Factor the function (e)

Start with \( f(x)=-x^2-4x + 21 \). Factor out - 1: \( f(x)=-(x^2 + 4x-21) \). Factor the quadratic inside the parentheses: \( x^2 + 4x-21=(x + 7)(x - 3) \), so \( f(x)=-(x + 7)(x - 3) \).

Answer:

a. Vertex: \((-2, 25)\)
b. y - intercept: \((0, 21)\)
c. The \(\text{maximum}\) value is \(25\)
d. x - intercept(s): \((-7, 0)\) and \((3, 0)\)
e. Factored form: \(f(x)=-(x + 7)(x - 3)\)