# GMAT Math : Calculating the equation of a curve

## Example Questions

### Example Question #1 : Calculating The Equation Of A Curve

Suppose the points  and  are plotted to connect a line. What are the -intercept and -intercept, respectively?

Explanation:

First, given the two points, find the equation of the line using the slope formula and the y-intercept equation.

Slope:

Write the slope-intercept formula.

Substitute a given point and the slope into the equation to find the y-intercept.

The y-intercept is: .

Substitiute the slope and the y-intercept into the slope-intercept form.

To find the x-intercept, substitute  and solve for x.

The x-intercept is:

### Example Question #2 : Calculating The Equation Of A Curve

Suppose the curve of a function is parabolic.  The -intercept is  and the vertex is the -intercept at .  What is a possible equation of the parabola, if it exists?

Explanation:

Write the standard form of the parabola.

Given the point , the y-intercept is -4, which indicates that .  This is also the vertex, so the vertex formula can allow writing an expression in terms of variables  and .

Write the vertex formula and substitute the known vertex given point  .

Using the values of , and the other given point , substitute these values to the standard form and solve for .

Substitute the values of ,, and  into the standard form of the parabola.

### Example Question #3 : Calculating The Equation Of A Curve

If the -intercept and the slope are , what's the equation of the line in standard form?

Explanation:

Write the slope intercept formula.

Convert the given x-intercept to a known point, which is .

Substitute the given slope and the point to solve for the y-intercept.

Substitute the slope and y-intercept into the slope-intercept formula.

Add 1 on both sides of the equation, and subtract  on both sides of the equation to find the equation in standard form.

### Example Question #4 : Calculating The Equation Of A Curve

Which of the following functions has as its graph a curve with , and  as its only two -intercepts?

Explanation:

By the Fundamental Theorem of Algebra, a polynomial equation of degree 3 must have three solutions, or roots, but one root can be a double root or triple root. Since the polynomial here has two roots,  and 4, one of these must be a double root. Since the leading term is , the equation must be

or

We rewrite both.

The correct response can be   or . The first is not among the choices, so the last is the correct choice.

### Example Question #1 : Calculating The Equation Of A Curve

Which of the following functions does not have as its graph a curve with  as an -intercept?

Explanation:

We can evaluate  in each of the definitions of  in the five choices. If  is an -intercept.

does not have  as an -intercept, so it is the correct choice.

### Example Question #2 : Calculating The Equation Of A Curve

A function is defined as

where are integer coefficients whose values (which might be positive, negative, or zero) are not given. Which of the following cannot be an -intercept of the graph of no matter what the values of those three coefficients are?

Explanation:

Since the graph of a function has its -intercept at a point if and only if , finding possible -intercepts of the graph of  is equivalent to finding a solution of . Since has integer coefficients, then by the Rational Zeroes Theorem, any rational solutions to the equation

must be the quotient, or the (negative) opposite of the quotient, of a factor of constant coefficient 12 - that is, an element of  - and a factor of leading coefficient 2 - that is, an element of . Since all of the choices are positive, we will only look at possible positive solutions.

The quotients of an element of the first set and an element of the last are:

;

;

Eliminating duplicates, the set of possible positive rational solutions to is

.

Of the five choices, only does not appear in the set of possible rational solutions of , so of the five choices, only cannot be an -intercept of the graph.

### Example Question #3 : Calculating The Equation Of A Curve

Between which two points is an -intercept of the graph of the function

located?

Between  and

Between  and

Between  and

Between  and

Between  and

Between  and

Explanation:

As a polynomial function,  has a continuous graph. By the Intermediate Value Theorem, if  and  are of different sign, then  for some  - that is, the graph of  has an -intercept between  and . Evaluate  for all  and observe between which two integers the sign changes.

Since  and , the -intercept is between  and .

### Example Question #4 : Calculating The Equation Of A Curve

Only one of the following equations has a graph with an -intercept between  and . Which one?

Explanation:

The Intermediate Value Theorem states that if  is a continuous function, as all five of the polynomial functions in the given choices are, and  and  are of different sign, then the graph of  has an -intercept on the interval .

We evaluate  and  for each of the five choices to find the one for which the two have different sign.

and  are both negative.

and  are both negative.

and  are of different sign.

and  are both positive.

and  are both positive.

is the function in which  and  are of different sign, so it is represented by a graph with an -intercept between  and . This is the correct choice.

### Example Question #5 : Calculating The Equation Of A Curve

Which of the following functions has as its graph a curve with -intercepts , , and  ?