Math 1314 College Algebra Final Review Solutions

Determine which point does not lie on the graph of the equation a.(1,6) b.(0,2) c.(-2,36) d.(-1,-8) e.(2,24) By substituting the x and y values of each point into the equation, you can determine whether the point is on the graph or not on the graph.

Determine the symmetry with respect to the axes and the origin. Replacing y by –y does not give the same equation, thus there is no symmetry with respect to the x-axis. Replacing x by –x does not give the same equation, thus there is no symmetry with respect to the y-axis. Replacing y by –y and x by –x does produce the same equation, thus there is symmetry with respect to the origin.

Write the standard form of the equation of the circle with the given characteristics. center: (-6, 4); solution point: (-7, 8)

Write the standard form of the equation of the circle with the given characteristics. Endpoints of the diameter: (5,-6), (9,-4) There are two things needed in order to write the standard form equation: center and radius. To find the center, you find the midpoint of the diameter. To find the radius, you find the distance from the center to one of the ends of the diameter.

Find the center and the radius of the circle

Solve the equation

Use the Quadratic Formula to solve

Solve the following quadratic equation.

The formula for the area of a rectangle is A = lw. Since the length is given in the diagram as w + 14, you can write the area as w(w + 14). Using the given area, you get the equation Solving the equation, which is a quadratic, you get: Since the width cannot be negative, w = 20 and w + 14 = 34. The length is 34 and the width 20.

Solve the equation and write complex solutions in standard form. It can be readily determined that this is not factorable since the factors of 25 do not add up to 8. You can use completing the square or the quadratic formula to solve the equation. Since the answer is to be in standard form, write this as two solutions.

Find all solutions to the equation

Find all solutions to the following equation. To solve a radical equation with one radical, isolate the radical, square both sides, and solve the resulting equation.

Find all solutions to the following equation.

Solve the equation: Begin by subtracting 8 from both sides of the equation.

Solve the inequality:

Solve the inequality.

Solve the quadratic inequality.

Solve the inequality.

Write the slope-intercept form of the equation of the line through the given point parallel to the given line. Now you can write the desired equation. Using the slope of the given line and the given point, substitute into the slope-intercept form equation

Write the slope-intercept form of the equation of the line through the given point perpendicular to the given line. From this equation you can find the slope. Since the problem is to write the equation of the line perpendicular to this one, use the negative reciprocal for the slope of the new line.

Evaluate the function at the specified value of the independent variable and simplify.

Find the domain of the function. There are two things that affect the domain, radicals and rational expressions. A radical cannot have a negative under the radical and a rational expression cannot have a 0 in the denominator. This problem is a rational function, so you must determine which values of x make the expression in the denominator 0. Write this in interval notation.

Find the domain of the function. There are two things that affect the domain, radicals and rational expressions. A radical cannot have a negative under the radical and a rational expression cannot have a 0 in the denominator. This problem is a radical function, so you must determine which values of x make the expression under the radical non-negative.

Find the difference quotient and simplify your answer.

Determine the interval on which the function in the graph below is increasing. As you follow the function from left to right, the first section is constant, the next section rises to the right, and the last section falls to the right. The only section that is increasing is the second. The x values range from 0 to 1.

The 4 inside the parentheses indicates a horizontal shift to the right 4. The 4 that is the coefficient indicates a stretching. The sequence of transformations is: Horizontal shift 4 units right then a vertical stretch by a factor of 4.

Write the function that is described by the following characteristics: To move a graph up, add to the function. To move the graph to the right, subtract from the variable.

Find the inverse function of f. The method used to find the inverse is to swap x and y and solve for y.

Find the vertex of the parabola. There are two methods for finding the vertex of a parabola. One method is to use completing the square. The other method is to find the x-coordinate of the vertex from a formula and then find the y-coordinate.

Write the standard form of the function of the parabola

Write the standard form of the function of the parabola that has a vertex at

From the diagram in the problem, you can determine that there are 3 fence segments of length x and 4 of length y. Since the farmer has 240 feet of fencing, you get the equation Solve this equation for either x or y. Thus The dimensions are: 30 ft. x 40 ft.

Describe the right-hand and the left-hand behavior of the graph of Since the degree of the equation is even, both the right-hand and left- hand behavior are the same. Since the lead coefficient is negative, the graph is upside down, which means the graph falls to the left and falls to the right.

Use the graph to determine the right-hand and left-hand behavior of the function. Since the graph rises to the left and rises to the right, the degree of the function must be even (even degree, both sides have the same behavior; odd degree, the sides have opposite behavior) and the lead coefficient must be positive (positive – up, negative – down). Looking at the functions, you can determine that the function for choice a meets the first term requirements for this graph.

The last number in the synthetic division process is the remainder. The remainder is 3.

List all possible rational zeros given by the Rational Zeros Theorem. Do not check to see which actually are zeros. The factors of 10 will be in the numerator and the factors of 3 will be in the denominator.

Find all real zeros of the polynomial There are two methods that can be used to find all the real zeros of this function. One method is to use synthetic division to find one zero and then find the remaining two zeros by solving the resulting quadratic. A second method is to see if the function is factorable. If the function is factorable, then factor and solve.

Zeros of a rational function are determined by the numerator. The domain and/or vertical asymptotes are determined by the denominator. To find the zeros, set the numerator equal to 0 and solve.

Determine the equations of any horizontal and vertical asymptotes of For the horizontal asymptote compare the degree of the numerator with the degree of the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the quotient of the terms with the highest degree. If the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote. This function meets the conditions of case 2. The horizontal asymptote is: For the vertical asymptote(s), set the denominator equal to 0 and solve. If the numerator is factorable, factor and make sure the numerator and denominator have no common factors. If they do, then the graph has a hole, not a vertical asymptote at that point.

Determine the equations of any horizontal and vertical asymptotes of Look at the pervious problem for an explanation of horizontal and vertical asymptotes.

Solve the equation: In order to solve an exponential equation, the term with the exponent must be isolated.

Solve the equation. This is a logarithmic equation. To solve, rewrite the left side as a single term. Rewrite the resulting logarithmic equation in exponential form. Solve the resulting equation. DO NOT FORGET TO CHECK.

This is an exponential equation. Make the bases the same.

Solve for x: Convert the log to exponential form and solve for x.

Solve the system. Since one of the equations has a squared term, use substitution. Solve the first equation for y and substitute.

Solve the system of equations for real values of ‘x’ only.

Solve the system. Use the method of elimination. Multiply the second equation by -9. Add the equations.

Solve the system. Multiply the first equation by 5. Since the equations are the same, which means they are dependent, there are infinitely many solutions.