Question 1

Solve the equation $$|4x + 12| = 20. $$What are the solutions for $$x$$?

Accepted Answer

$$x = 2,\ x = -8$$

Question 2

Find the domain of the function $$f(x) = \frac{1}{\sqrt{2x + 10}}$$ and write your answer in interval notation.

Accepted Answer

$$(-5,\ \infty)$$

Question 3

Solve $$x^2 + 10x = 11$ by completing the square.

Accepted Answer

$$x = -5 \pm 2\sqrt{6}$$

Question 4

Solve the quadratic inequality $$x^2 - 2x$$ \geq 15$$ and write the solution in interval notation.

Accepted Answer

$$(-\infty,\ -3]\ \cup\ [5,\ \infty)$$

Question 5

Solve $$3x^2 + 75 = 0$ and provide the answer in a + bi$ form.

Accepted Answer

$$x = 0 \pm 5\sqrt{3}i$$

Question 6

Calculate the distance between the points $$(1, 5)$$ and $$(-3, 2)$$ using the distance formula.

Accepted Answer

$$5$$

Question 7

A piece of string with length $$20x is $$formed into a circle. Express the area $$A of $$the circle as a function of $$x.$$

Accepted Answer

$$A(x) = \frac{100x^2$}{\pi}$$

Question 8

For the function $$h(x) = -2(x + 3)^2$ - 5$$, which of the following lists all the transformations applied to the parent function $$y = x^2$?

Accepted Answer

Shift left 3 units, reflect over $$x-$$axis, vertical stretch by 2, shift down 5 units

Question 9

What is the domain and range of $$h(x) = -2(x + 3)^2$ - 5$$?

Accepted Answer

Domain: $$(-\infty,\ \infty)$$; Range: $$(-\infty,\ -5]$$

Question 10

For the quadratic function $$f(x) = x^2$ - 6x + 5$$, what is the vertex and the equation for the axis of symmetry?

Accepted Answer

Vertex: $$(3, -4)$$; Axis: $$x = 3$$

Question 11

For $$f(x) = x^2$ - 6x + 5$$, does the parabola open up or down, and what are the $$x-$$intercepts?

Accepted Answer

Opens up; $$x-$$intercepts at $$x = 1$$ and $$x = 5$$

Question 12

Given the system of equations:
$$2x + y = 10$$
$$x - y = 2$$
Solve for $$x$$ and $$y$$ using the substitution method.

Accepted Answer

$$x = 4,\ y = 2$$

Question 13

Solve the system using elimination and state if it is inconsistent:
$$4x - 2y = 8$$
$$8x - 4y = 10$$

Accepted Answer

The system is inconsistent; no solution

Question 14

Which of the following is the correct formula for the distance between points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$d = \sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Question 15

Given $$g(x) = a(x - h)^2$ + k$$, which parameter controls the horizontal shift and in which direction does $$h \u003e 0$$ shift the graph?

Accepted Answer

$$h$$; right

Question 16

Which of the following describes the range of a function $$f(x)$$ whose graph opens downward and has a vertex at $$(2, 7)$$?

Accepted Answer

$$(-\infty,\ 7]$$

Question 17

If a function $$f(x) is $$decreasing on the interval $$(-\infty, 0)$$ and increasing on $$(0, \infty)$$, what can you say about $$x = 0$$?

Accepted Answer

$$x = 0 is a $$local minimum

Precalculus Midterm Review: Equations, Functions, Quadratics, and Systems

Study Guide - Practice Questions