Question

17
use the drawing tools to form the correct answer on the graph.
draw a line for the axis of symmetry of function f. also mark the x-intercept(s), y-intercept, and vertex of the function.
$f(x) = x^2 - 4x - 5$
drawing tools interface and graph omitted

Explanation:

Step1: Find the axis of symmetry

For a quadratic function \( f(x) = ax^2 + bx + c \), the axis of symmetry is given by \( x = -\frac{b}{2a} \). For \( f(x) = x^2 - 4x - 5 \), \( a = 1 \), \( b = -4 \). So, \( x = -\frac{-4}{2\times1} = 2 \). The axis of symmetry is the vertical line \( x = 2 \).

Step2: Find the vertex

The x - coordinate of the vertex is the same as the axis of symmetry, \( x = 2 \). Substitute \( x = 2 \) into the function: \( f(2) = 2^2 - 4\times2 - 5 = 4 - 8 - 5 = -9 \). So the vertex is \( (2, -9) \).

Step3: Find the x - intercepts

Set \( f(x) = 0 \), so \( x^2 - 4x - 5 = 0 \). Factor the quadratic: \( (x - 5)(x + 1) = 0 \). This gives \( x - 5 = 0 \) or \( x + 1 = 0 \), so \( x = 5 \) or \( x = -1 \). The x - intercepts are \( (-1, 0) \) and \( (5, 0) \).

Step4: Find the y - intercept

Set \( x = 0 \) in the function: \( f(0) = 0^2 - 4\times0 - 5 = -5 \). So the y - intercept is \( (0, -5) \).

To draw on the graph:

• Draw the vertical line \( x = 2 \) for the axis of symmetry.
• Mark the points \( (-1, 0) \), \( (5, 0) \) as x - intercepts, \( (0, -5) \) as y - intercept, and \( (2, -9) \) as the vertex.

Answer:

• Axis of symmetry: Vertical line \( x = 2 \)
• Vertex: \( (2, -9) \)
• x - intercepts: \( (-1, 0) \), \( (5, 0) \)
• y - intercept: \( (0, -5) \) (And draw the axis of symmetry \( x = 2 \) and mark the above - mentioned points on the given graph)