College Algebra Problems With Answers - sample 1: Quadratic Functions

College Algebra Problems With Answers
Sample 1 : Quadratic Functions

A set of college algebra problems on quadratic functions, with answers, are presented. The solutions are at the bottom of the page.

Problem 1

Write the quadratic function f(x) = -2x2 -12x - 20 in standard form (or vertex form).

Problem 2

Let f(x) = - 2 x2 + 4x + 6.
a) Find the vertex of the graph of f.
b) Find the range
of f.
c) Find the x - intercepts of the graph of f.
c) Find the y - intercept of the graph of f.

Problem 3

f is a quadratic function whose graph has a vertex at the point (-3 , 2) and has a y-intercept at the point (0 , -16).
a) Find the equation for function f.
b) Find the x-intercepts of the graph of f.

Problem 4

Find all values of b and c so that the quadratic function f(x) = x2 + bx + c has a graph that is tangent to the x axis and a y-intercept at (0 , 4).

Problem 5

Find the equation of a quadratic function f whose graph has a vertical axis of symmetry x = -2, the range of f is given by the interval [4 , +infinity) and f(2) = 8.

Problem 6

A graph of a parabola is shown below. Find
a) The vertex of the parabola. (round the x and y coordinates to the nearest integer).
b) The x and y intercepts of the parabola. (round the x and y coordinates to the nearest integer).
c) The equation of the axis of symmetry of the parabola.
d) The equation of the line tangent to the parabola at the vertex.
e) The equation of parabola.
college algebra graph of parabola in problem 6 .
Figure 1. graph of parabola in problem 6

Problem 7

A rectangular picture measuring 10 cm by 15 cm is surrounded by a frame with uniform width x. Write a quadratic function that gives the area A of the frame as a function of x.

Problem 8

Function g is given by g(x) = 2(x - 3)(x- 7). Find the value of x that makes g(x) maximum.

Solutions to the Above Problems

Solutions to Problem 1

f(x) = -2x2 -12x - 20 = -2(x + 3)2 - 2 (completing square)

Solutions to Problem 2

  1. (1 , 8)
  2. (-infinity , 8]
  3. (-1 , 0) , (3 , 0)
  4. (0 , 6)

Solutions to Problem 3

  1. f(x) = -2(x + 3)2 + 2
  2. (-2 , 0) , (-4 , 0)

Solutions to Problem 4

c = 4 , b = 4 or -4

Solutions to Problem 5

f(x) = (1/4)(x + 3)2 + 4

Solutions to Problem 6

  1. (-2 , 4)
  2. x-intercept: (-4 , 0) , (0 , 0) y-intercept:(0 , 0)
  3. x = -2
  4. y = 4
  5. y = -(x + 2)2 + 4

Solutions to Problem 7

A = (10 + 2x)(15 + 2x) - 10*15 = 4x2 + 50x

Solutions to Problem 8

g(x) is maximum at x = 10

More References and links

Vertex and Intercepts Parabola Problems
Find Range of Quadratic Functions
How to Find x and y Intercepts Of Graphs
quadratic functionAlgebra Questions and problems
More ACT, SAT and Compass practice