Part A:
To find the x-intercepts of the graph of f(x), we need to set f(x) = 0 and solve for x:
4x² + 8x - 5 = 0
We can use the quadratic formula to solve for x:
x = (-b ± sqrt(b² - 4ac)) / 2a
where a = 4, b = 8, and c = -5.
x = (-8 ± sqrt(8² - 4(4)(-5))) / 2(4)
x = (-8 ± sqrt(144)) / 8
x = (-8 ± 12) / 8
x = -1/2 or x = 5/2
Therefore, the x-intercepts of the graph of f(x) are -1/2 and 5/2.
Part B:
The vertex of the graph of f(x) can be found using the formula:
x = -b / 2a
y = f(x)
where a = 4, b = 8, and c = -5.
x = -8 / 2(4)
x = -1
y = f(-1)
y = 4(-1)² + 8(-1) - 5
y = -1
Therefore, the vertex of the graph of f(x) is (-1, -1). Since the coefficient of the x² term is positive, the parabola opens upwards, so the vertex is a minimum.
Part C:
To graph f(x), we can use the information obtained in Part A and Part B. The x-intercepts are -1/2 and 5/2, and the vertex is (-1, -1). We can also find the y-intercept by setting x = 0:
f(0) = 4(0)² + 8(0) - 5
f(0) = -5
Therefore, the y-intercept is (0, -5).
We can also find the axis of symmetry by using the x-coordinate of the vertex:
x = -1
Therefore, the axis of symmetry is x = -1.
To draw the graph, we can plot the x-intercepts, y-intercept, and vertex, and then sketch a smooth curve passing through these points. Since the vertex is a minimum, the curve will open upwards. We can also use the axis of symmetry to help us draw the curve symmetrically.
Therefore, the steps to graph f(x) are:
1. Find the x-intercepts by solving 4x² + 8x - 5 = 0.
2. Find the y-intercept by setting x = 0.
3. Find the vertex by using x = -b / 2a and y = f(x).
4. Find the axis of symmetry by using the x-coordinate of the vertex.
5. Plot the x-intercepts, y-intercept, and vertex.
6. Sketch a smooth curve passing through these points, opening upwards and symmetrically about the axis of symmetry.