# Points, Vectors, and Functions

### Topics

In three easy steps, we solve any math problem. And for three easy payments of...

We're not trying to sell you a set of knives that will never dull or a counter top oven that'll cook eggs, bacon and toast all at once. We can at least sell the basic steps to solving every math problem.

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

### Sample Problem

Determine all points at which the graph of the parametric function *x* = -5 + 5*t y* = 4*t *and the graph of the parametric function *x* = 2*t* + 1 *y *= *t*^{2 }intersect. Round to 5 digits, use a calculator.

Answer.

- Figure out what the problem is asking.

The problem is asking for the coordinates of all points (*x*,*y*) that occur on both graphs.

**Important:** If (*x*,*y*) occurs on both graphs, it will probably result from different values of the parameter for each graph.

We might want to set the *x* equations equal and solve

-5 + 5*t* = 2*t* + 1 for *t*, but that will only work if the same value of *t* generates the point for both graphs. That probably isn't the case.

Instead, an intersection point occurs when we put some value *t* into the first set of equations and some value *s* into the second set of equations and find the same point (*x*,*y*).

Writing this in symbols, we want

*x* = -5 + 5*t* = 2*s* + 1 and *y* = 4*t* = *s*^{2}.

Dropping the *x* and *y*, we want -5 + 5*t* = 2*s* + 1 and 4*t* = *s*^{2}.

This is a system of two equations with two unknowns, which we know how to solve.

After finding *s* and *t* we need to find the coordinates of the point or points (*x*,*y*) that occur on both graphs.

- Solve the problem.

We need to solve the system

-5 + 5*t* = 2*s* + 1

4*t* = *s*^{2}.

Solve the second equation for *t*:

t = 1 / 4 s^{2}.

Then substitute for *t* in the first equation:

-5 + 5*t* = 2*s* + 1

-5 + 5(1/4)s^{2} = 2s + 1

This is a quadratic equation. Tidy it so we can use the quadratic formula to solve it.

-5 + 5(1/4)s^{2}) = 2s + 1

-6 + 5/4s^{2} = 2s

5\4 s^{2}- 2s - 6 = 0

5s^{2 }- 8s - 24 = 0

We didn't need to do the last step, where we multiplied through by 4, but it's easier to use the quadratic formula when the coefficients are integers instead of fractions.

The quadratic formula says

s = (8± sqrt{(-8)^{2}-4(5)(-24)}}{2(5)}

= (8 ± sqrt{544})\10.

There are two values of *s* here that we need to use. To find the appropriate values of *x* and *y* we use the second set of equations. When

s = (8 + sqrt{544})/10

we have

x = 2(8 + sqrt{544}/10) + 1

≅7.26476

and

y = (8 + sqrt{544})/10)^{2}

≅ 9.81181.

One point of intersection is approximately

(7.26476, 9.81181).

When

s = (8 - sqrt{544})/10

we have

x = 2(8 - sqrt{544}/10) + 1

≅ -2.06476

and

y = (8 - sqrt{544}/10)^{2}

≅ 2.34819.

The other point of intersection is approximately

(-2.06476, 2.34819).

- Check the answer.

To start with, graph the two sets of equations.

It looks like they run into each other twice, once somewhere around (-2,2) and once somewhere around (7,10).

This tells us that we found the correct number of points, and the numbers we found seem to be about right. In practice, this is probably all the checking we would do.

Now we'll show how to check the answers for real.

We'll find the value of *t* that corresponds to each value of *s*, and make sure that we find the correct point when we put that value of *t* into the first set of parametric equations.

When

s = (8 + sqrt{544})/10

we have

t = 1/4 s^{2}

= (1/4)(8 + sqrt{544}/10)^{2}.

Then

x = -5 + 5(1/4)(8 + sqrt{544}/10)^{2}

≅7.26476

and

y = 4(1/4)(8 + sqrt{544}/10)^{2}

≈ 9.81181

This agrees with the (*x*,*y*) coordinates we got from the second set of parametric equations, therefore we believe

(7.26476, 9.81181 )

is an intersection point.

When

s = (8 - sqrt{544})(10)

we have

t = (1/4)s^{2}

= 1/4(8 - sqrt{544}/10)^{2}.

Then

x = -5 + 5(1/4)(8 - sqrt{544})/10)^{2}

≈ -2.06476

and

y = 4(1/4)(8 - sqrt{544})/10)^{2}

≈ 2.34819

This agrees with the (*x*,*y*) coordinates we got from the second set of parametric equations, we believe

(-2.06476,2.34819)

is an intersection point.