Hi all -nice to meet you!
My grandson has this question (I must tell that my math was very poor 50 years ago)

2 functions:

1. y=-a^2+2a
2. y=5ax

• for which values of a there is only one "cut point" for the functions graphs ( ....I am trying to translate it maybe it is not the exact term!)
• for which a the functions have infinite cutpoints?
• for which a values -there are no cutpoints between function graphs?
Hope it make sense - as I know you mathematitions I am sure it make sense.

I will be grateful if you may also explain the logic of solving such problems!

Elie

2. Re: pls help solve my grandson's questions in an exam

Hello, eliW!

I hope I have interpreted the problem correctly.

Two functions: . $\begin{Bmatrix}y&=&-a^2+2a & [1] \\ y&=&5ax & [2]\end{Bmatrix}$

For which values of a is there only one intersection?
For which values of a are there infinite intersections?
For which values of a are there no intersections?

Function [1] is a horizontal line.
Function [2] is a line through the origin.

If $a = 0$, both graphs are the x-axis.
. . There are infinite intersections.

If $a \ne 0$, there is one intersection.

3. Re: pls help solve my grandson's questions in an exam

Hi Soroban and thank you for your correct interpratation and also the answer !

I can understand your solution just from looking - but I think that there must be some systematic way to have the solution e.g. by manipulating the 2 equations? so a schoolboy of ninth grade can solve it by some method.

Also how can you show that thereis no a of which the 2 graphs have no intersections at all.

Is it enough to say tha 2 linear functions can have only one or infinite intersections?

All the best,

Elie

4. Re: pls help solve my grandson's questions in an exam

What do you mean by "questions in an exam"? Is this work going to count towards his final grade?

5. Re: pls help solve my grandson's questions in an exam

oh sorry -I made a mistke in copying the 2 functions
So here is the question:
Two functions:
[1] y = -a^2*x + 2a
[2] y = 5ax

For which values of a is there only one intersection?
For which values of a are there infinite intersections?
For which values of a are there no intersections?

Quacky: my grandson had this exam in class and he failed to answer this one question. that is all. I am trying from time to time to help him - and btw I enjoy it very much. Can you assist ?
Elie

6. Re: pls help solve my grandson's questions in an exam

We have:
$y={\color{red}-a^2}x+2a$
$y={\color{red}5a}x$

The red is just there to show the corresponding parts of the equation. As you may be aware, the red parts tell us the gradient, ie the "slope", "direction" or "angle" of the lines.

Consider the case where $-a^2=5a$, in which case, the gradient of the lines (the parts in red) are equal. If we add $a^2$ to both sides of this equation, we get that $0=5a+a^2$. As I'm sure your grandson will be quick to point out, this factors into $a(5+a)=0$, which can be solved to give $a=-5$ and $a=0$.

Task: consider each of these cases with the original equations. For example, when $a=-5$, the first equation becomes $y=-25x-10$ and the second equation becomes $y=-25x$. These lines have the same gradient, but clearly they are not the same line - a visual representation can be seen by clicking here. Clearly, the lines have the same gradient - they're parallel - and therefore they will never intersect.

Can you (or your grandson) consider the case when $a=0$?

Edit: Corrected a minor typo. The link has been updated to reflect this.

7. Re: pls help solve my grandson's questions in an exam

Wonderful !!!

So can we say that besides a=-5 in which there are no intersections
and a=0 in which there are infinite intersections
in all other cases {say AND(a<>-5,<>0)} - there is 1 intersection???

BTW how do you get the mathematical fonts?
And also wht this is called quadratic questions?
Thanyou very much.