Final answer:
To graph the quadratic function f(x), neither of the given factorizations match the original function. We must find the x-intercepts, y-intercept, and vertex, and use the end behavior to draw the graph, which opens upwards due to the positive leading coefficient.
Step-by-step explanation:
To answer these questions, we'll first need to verify whether the given function f(x) = 4x² + 8x - 5 can be factored into one of the provided options (a, b, c, or d).
Part A: By applying the factoring techniques, none of the given options exactly match the factorization of f(x). Therefore, we need to use the quadratic formula or complete the square to factor it if it is factorable.
Part B: The x-intercepts of f(x) are the values of x that make f(x) = 0. We can find this by setting our factored form, if available, equal to zero and solving for x.
Part C: The end behavior of f(x) is determined by the leading coefficient and the highest power of x. As it is a quadratic function with a positive leading coefficient, as x approaches infinity or negative infinity, f(x) will also approach infinity. This is because the function opens upwards.
Part D: To graph f(x), follow these steps:
- Find the x-intercepts by setting f(x) equal to zero and solving for x.
- Find the y-intercept by evaluating f(0).
- Find the vertex by using -b/(2a) to find the x-coordinate and then evaluate f(x) at that point to find the y-coordinate.
- Plot the x-intercepts, y-intercept, and the vertex on a coordinate plane.
- Sketch the parabola, using the vertex as the turning point and ensuring that the end behavior follows the quadratic's nature.
The answers obtained in Part B and Part C are essential to drawing the graph as they give us specific points where the graph will intersect the x-axis and information about the direction the graph takes as it extends towards infinity.