Answer:
(a) x-intercepts: (-4, 0) and (1, 0)
y-intercept: (0, 4)
(b) Axis of symmetry: x = -1.5
(c) Vertex: (-1.5, 6.25)
(d) See attachment
Step-by-step explanation:
Part (a)
The x-intercepts are the points at which the graph intersects the x-axis, so where the function f(x) equals zero.
Set f(x) = 0:
[tex]-(x + 4)(x - 1) = 0[/tex]
This gives us two solutions:
[tex]x + 4 = 0 \implies x = -4\\\\ x - 1 = 0 \implies x = 1[/tex]
So, the x-intercepts are (-4, 0) and (1, 0).
The y-intercept is the point at which the graph intersects the y-axis, so where x = 0.
Substitute x = 0 into f(x):
[tex]f(0) = -(0 + 4)(0 - 1) \\\\f(0)= -4(-1) \\\\f(0)= 4[/tex]
So, the y-intercept is (0, 4).
[tex]\dotfill[/tex]
Part (b)
The axis of symmetry is the midpoint of the x-coordinates of the x-intercepts. Given that the x-intercepts are (-4, 0) and (1, 0):
[tex]x = \dfrac{-4 + 1}{2} \\\\\\x= \dfrac{-3}{2} \\\\\\x= -1.5[/tex]
So, the axis of symmetry is x = -1.5.
[tex]\dotfill[/tex]
Part (c)
The vertex lies on the axis of symmetry, so its x-coordinate is x = -1.5. To find the y-coordinate of the vertex, substitute x = -1.5 into the function f(x):
[tex]f(-1.5) = -(-1.5 + 4)(-1.5 - 1)\\\\f(-1.5) =-(2.5)(-2.5)\\\\f(-1.5)=6.25[/tex]
So, the vertex is (-1.5, -6.25).
[tex]\dotfill[/tex]
Part (d)
The parabola opens downwards because the leading coefficient of the quadratic term is negative.
To graph the function:
- Plot the x-intercepts (-4, 0) and (1, 0), the y-intercept (0, 4), and the vertex (-1.5, 6.25).
- Draw the axis of symmetry as a dashed line at x = -1.5.
- Sketch a downward-opening parabola that passes through the plotted points and is symmetric about the axis of symmetry.
