Math Help - Afew Questions

Any help with the following questions is appreciated

Linear/Non-Linear Systems of Equations

1. Write an equation to form a system with so that the system has (2,9) as its only solution.

Answer is
But how do i get that answer?

2. The range of the graph is: ?



Quadratic Functions

1. What is the new equation when is translated 5 units left and 3 units down?


Quadratic Equations

1. Determine the values of K so that the graph crosses the x-axis twice.







Im wondering how to tell if i need a greater/less or equal sign (I think it is > for 2 real roots, < 2 unreal, = roots that are equal) also why it switched on the last step from -4k>-16 to K<4

Polynomial Functions/Equations

1. What are the zeroes of ?

I got
But i think there are others?

I must ask you what in the world that posting means?
It seems to be just series of meaningless statements.
What are you on about?

Was in a rush sort of, i can't believe i forgot my notes at school >.<

I think its more clearer now.

Originally Posted by Raj

1. Write an equation to form a system with so that the system has (2,9) as its only solution.

Answer is
But how do i get that answer?

I presume that you are looking for another linear function, else there are an infinite number of answers.

So let's assume we have a new equation of the form .

So solve the system


such that the solution is (2, 9).

I'll start you off:
Put the solution point into the second equation:


So


Thus the second equation is


Now we need to solve the system


for m.

See what you can do with this.

-Dan

Originally Posted by Raj

2. The range of the graph is: ?

The range of a graph is the set of all possible y values that the function takes. So what y values can this function take?

(Hint: Graph it.)

-Dan

Originally Posted by Raj

1. What is the new equation when is translated 5 units left and 3 units down?

Consider the function .
When we translate this graph h units to the right the corresponding function is .

When we translate this graph h units to the left the corresponding function is .

When we translate this graph k units up the corresponding function is .

When we translate this graph k units down the corresponding function is .

See if you can apply these rules to get your answer.

-Dan

Originally Posted by Raj

1. Determine the values of K so that the graph crosses the x-axis twice.







Im wondering how to tell if i need a greater/less or equal sign (I think it is > for 2 real roots, < 2 unreal, = roots that are equal) also why it switched on the last step from -4k>-16 to K<4

Note that if the discriminant is equal to 0 then there is only one solution to the quadratic equation. Thus we can't include the equal sign on the line .

Whenever you divide both sides of an inequality by a negative number the inequality changes "direction." So






If you have troubles seeing this, consider a simpler example. Solve


You can easily get the solution set from the given inequality, and you should be able to see that it is , which is what you get when you divide both sides by -1.

-Dan

Originally Posted by Raj

1. What are the zeroes of ?

I got
But i think there are others?

I presume this is really ?

Why should there be other zeros?

The concept behind solving something like this is the following theorem:
If ab = 0 then either a = 0 or b = 0 or both.

So when we a simpler example like

then either

or

(and we can't have both terms equal to 0 at the same time.) You've likely seen this process any number of times when solving a quadratic that factors.

In this case we have:




So either
<-- Not true!
or

or

or

or


So we have


or more simply


-Dan