Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Quadratic Function

DEFINITION.
A quadratic function is a polynomial of degree 2:



The graph is a parabola.
• If a > 0, the horns point up.

If a < 0, the horns point down.

If |a| > 1, the parabola is narrower than y = x2.

If |a| < 1, the parabola is wider than y = x2.

Determine the shape of the following graphs. Pick the
shape below.

Given
Get the roots by factoring or using the quadratic formula:
. No roots if

COMPLETING THE SQUARE THEOREM. Every quadratic
function may be written in the form:



where is the vertex (or nose) of the parabola.

Proof. Given

• Factor the a out of the ax2+bx part.

• Divide the new coefficient of x by 2 and square.
Add this to complete the square.

• Anything which is added must also be subtracted to
preserve equality.

• Find the roots (the roots are the x-intercepts).

• Write in completed square form:

• Graph. On the graph list both coordinates of the vertex.


Find the vertex.

Find the graph with the correct shape and position.

Find the roots. x = ?, ?
Write equation in the form:

Find the vertex.
 

WORD PROBLEMS

• Draw the picture. Indicate the variables in the picture.
• Write the given equations which relate the variables.
• Solve for the wanted quantities.

The perimeter of a rectangle is 10 feet.
Express the area A in terms of the width x.

The area of a rectangle is 10 square feet.
Express the perimeter P in terms of the width x.

List the given.

Write the perimeter as a function of x

The corner of a triangle lies on the line

Express the triangle’s area and perimeter in terms of the
base x.

The area of an isosceles triangle is 16.

Write the triangle’s height h in terms of its width w.

Write the triangle’s width w in terms of its height h.

The curved surface
area is the area of the
can’s side, excluding
the top and bottom

The height of a can (right circular cylinder)
is three times the radius.
Express the radius as a function
of the curved surface area.

The height of a can (right circular cylinder)
is four times the radius.
Express the curved surface area as a function
of the radius.