I understand that my guide is kinda helpful, so if you find it difficult to go to my Facebook wall, you can just come here and see, and I hope it helps, somehow.
Cheers,
~LazyPika
Idiots Guide to : Unit 3 - Quadratic Graphs
Note : This guide is intended for Secondary 2 students on the topic of Quadratic Graphs, but if you are not in the category of the said intended audience, you are still welcome to read, as it may help you somehow.
The equation for Quadratic Graphs is y = ax²+bx+c (In Quadratic Equations [Unit 2], y is always 0)
Quadratic Graphs are always in the form of parabolas.
There are 2 formulas for Quadratic Graphs, namely completing the square and cross factorisation.
Completing the square --- y = (x-p)²+q
Cross factorisation --- y = (x-a)(x-b)
Completing the square
In completing the square, the formula is y = (x-p)²+q. [NOTE : It is -p, not +p]
To find out how to draw the graph itself, follow these few steps.
- Find the turning point
- Find the line of symmetry
- Find the y-intercept
- Find the x-intercept(s) [if any]
#1 - Turning point
The turning point is the lowest point of a U-shaped graph and the highest point of an N-shaped graph. As it is a point in the graph, it will always have an x- and a y-coordinate.
In the formula y = (x-p)²+q, the turning point is (p,q).
#2 - Line of symmetry
The line of symmetry is a line that intersects the graph and splits it exactly in half. The line of symmetry is located along the x-axis.
In the formula y = (x-p)²+q, the line of symmetry is x = p.
#3 - Finding the y-intercept
The y-intercept is the point at which the graph intercepts the y-axis. The y-intercept is located along the y-axis.
To find the y-intercept, simply substitute x for 0, and proceed to solve the equation. It is always in the form of (0,y).
#4 - Finding the x-intercept(s) [if any]
The x-intercept(s) is/are the point(s) at which the graph intercepts the x-axis. The x-intercept(s) is/are located along the x-axis.
To find the x-intercept(s), simply substitute y for 0, and proceed to solve the equation. They are always in the form of (x,0).
Note that depending on the coefficient of x², and the values of p and q, there may be two, one or even no x-intercepts.
In the formula y = (x-p)²+q, when y = 0 and the coefficient of x² is 1 :
When q > 0, there are no x-intercepts.
When q = 0, there is only one x-intercept (the turning point of the graph).
When q < 0, there are two x-intercepts.
In the formula y =-(x-p)²+q, when y = 0 and the coefficient of x² is -1 :
When q > 0, there are two x-intercepts.
When q = 0, there is only one x-intercept (the turning point of the graph).
When q < 0, there are no x-intercepts.
[Note that as the formula involves square-rooting a number, it may turn out as an irrational number, and in such cases you should always round off the values to 3 significant figures, unless specifically stated otherwise.]
Special case(s) :
If the equation is, for example, x²+3, it is equal to (x+0)²+3 (thus p = 0, q = 3).
If the equation is, for example, (x-5)², it is equal to (x-5)²+0 (thus p = 5, q = 0).
After you find the turning point, the line of symmetry, and the x- and y-intercepts, you can now draw the graph.
Cross-factorisation
In completing the square, the formula is y = (x-a)(x-b). [NOTE : It is -a and -b, not +a and +b]
To find out how to draw the graph itself, follow these few steps.
- Find your x-intercepts
- Line of symmetry
- Turning point + y-intercept
#1 - Find your x-intercepts
The x-intercept(s) is/are the point(s) at which the graph intercepts the x-axis. The x-intercept(s) is/are located along the x-axis. They are always in the form of (x,0).
The x-intercepts are (a,0) and (b,0).
#2 - Line of symmetry
The line of symmetry is a line that intersects the graph and splits it exactly in half. The line of symmetry is located along the x-axis.
In the formula y = (x-a)(x-b), the line of symmetry is (a+b)/2.
#3 - Turning point + y-intercept
The turning point is the lowest point of a U-shaped graph and the highest point of an N-shaped graph. As it is a point in the graph, it will always have an x- and a y-coordinate.
In the formula y = (x-a)(x-b), the turning point is ([(a+b)/2], y).
To find the y-intercept, substitute x for 0, and solve the equation.
(Make the formula y = ([(a+b)/2]-a)([(a+b)/2]-b).)
After you find the turning point, the line of symmetry, and the x- and y-intercepts, you can now draw the graph.
Follow these steps to draw your graph.
- Draw the axises
- Locate the turning point
- Plot your graph
- Label your graph
#1 - Draw your x- and y-axis
You don't need me to tell you how to draw this. Don't forget to label both axises and the origin (just put a 0 there).
#2 - Locate the turning point
On your axises, label the co-ordinates for the turning point and put a little dot on the spot where the the turning point should be.
#3 - Plot your graph
Draw a curve as natural-looking as humanly possible. Try to make your x- and y-intercepts look realistic. (Like, don't put 1 and 568 so close together on the x-axis then 1 and 3 so far apart on the y-axis, get what I mean?)
#4 - Label your graph
Make sure you :
Label the dot, your turning point (p,q)
Draw your line of symmetry and label it(x = p) [need not be dotted, this is not physics]
Label the y-intercept and x-intercept(s) [if any ; when applicable]
Label your graph [In its original form y = (x-p)²+q or y = (x-a)(x-b).]
And that is how you should draw your quadratic graph. Remember that you should always read the question first before assuming anything, and use a pencil so you can erase if you make a mistake. Thanks for your kind attention! ^~^
Note : This tutorial is self-written, adapted from Ms Ng Ee Nee's teaching. Thus it may differ slightly from the teachings of Ms Ng Ee Nee, Mr Thomas Yeo or Ms Gwendolyn Lim, depending on who your Mathematics teacher is.
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