You will need to revise the meanings of the equations of each type of graph. And also think about how y will change as you increase or decrease x. And, whether it is negative or positive.

For example, (b) is x².

Why?

For any real value of x, x² is always equal to or bigger than 0

x² ≥ 0

The square of a negative value is positive (remember the two negatives cancel out each other. Eg. (-5)² = (-5) × (-5) = 25)

The square of 0 is still 0.

The square of a positive is still positive
(eg. 3² = 3 × 3 = 9)


So for y = x², when x = 0, y = 0.
When y gets more positive or negative, x will only get more positive. And it will only increase more and more quickly and greatly.

(-5)² = 25
(-4)² = 16
(-3)² = 9
(-2)² = 4
(-1)² = 1
0² = 0
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25

And so on.

This explains the shape of the parabola, a quadratic curve.

(c) is y = 5x²

The shape is the same as (b), but narrower.

Why? Because now there is a multiple of 5 in front of x².


So as x increases, y will increase even faster. So it takes a much smaller increase in x to get a same value of y as the other graph y = x².

5(1²) = 5
5(2²) = 20
5(3²) = 45


What's common about (b) and (c) is that, the y-values are never negative. Minimum 0.

(e) is y = -4x².

Reason :

Since x² ≥ 0 for all real values of x,

Then -4x² is always ≤ 0 for all real values of x.

The negative sign in front results in this.


So the graph has a maximum y-value of 0. It only gets more negative when x is further away from 0.


We normally describe y = -x² as a reflection of the graph y = x² about the x-axis.

They are mirror images of each other.

As for (a) and (d), consider this :

Q : What is the minimum point (y value) ?
A : 0.


Q : When does y equal 0 ?
A : When y = 3(x + 2)² = 0 and y = 3(x - 2)² = 0


Q : what is the value of x such that x + 2 = 0 and x - 2 = 0?
A : -2 and 2 respectively.


Q : So what are the x-values of the minimum points of the graphs?
A : x = -2 and x = 2

Thank u so much for ur time and effort really helped:)

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