# 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.
## Problems

### Problem 1

Write the quadratic function \( f(x) = -2x^2 -12x - 20 \) in standard form (or vertex form).### Problem 2

Let \( f(x) = - 2 x^2 + 4x + 6 \).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 \( f(x) = x^2 + bx + c \) 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 \([4 , +\infty)\) and \( f(2) = 8 \).### Problem 6

A graph of a parabola is shown below. Finda) 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.

### 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

\( f(x) = -2x^2 -12x - 20 = -2(x + 3)^2 - 2 \) (completing square)### Solutions to Problem 2

- \( (1 , 8) \)
- \( (-\infty , 8] \)
- \( (-1 , 0) , (3 , 0) \)
- \( (0 , 6) \)

### Solutions to Problem 3

- \( f(x) = -2(x + 3)^2 + 2 \)
- (-2 , 0) , (-4 , 0)

### Solutions to Problem 4

\( c = 4 \) , \( b = 4 \) or \( -4 \)### Solutions to Problem 5

\( f(x) = \frac{1}{4}(x + 3)^2 + 4 \)### Solutions to Problem 6

- (-2 , 4)
- x-intercept: (-4 , 0) , (0 , 0) y-intercept:(0 , 0)
- \( x = -2 \)
- \( y = 4 \)
- \( y = -(x + 2)^2 + 4 \)

### Solutions to Problem 7

\( A = (10 + 2x)(15 + 2x) - 10 \times 15 = 4x^2 + 50x \)### Solutions to Problem 8

\( g(x) \) is maximum at \( x = 10 \)## More References and links

Vertex and Intercepts Parabola ProblemsFind Range of Quadratic Functions

How to Find x and y Intercepts Of Graphs

quadratic function Algebra Questions and problems

More ACT, SAT and Compass practice