Graph of Quadratic Functions
(1) Of the form y= ax2 + bx + c ( b and c can be zero, but a ≠ 0)
(2) All have a line of symmetry
(3) U-shaped
(4) Turning Point:
(i) Positive [+ax2] graph - turning point at the bottom (Minimum point)
(ii) Negative [-ax2] graph - turning points at the top (maximum)
Plotting the Graph
Example (1)
(a) Complete the table for y = -x2 + x -3 for -2 <= x <=3.
X
|
-2
|
-1
|
0
|
1
|
2
|
3
|
Y
|
-9
|
(b) Plot and draw the graph.
(c) Use the graph to solve the equations
(a) –x2 + x – 3 = -7
(b) –x2 + x – 3 = 0
(a) Filling in the table
(a) The table is for the x and y values of equation y = –x2 + x – 3
Step 1: Write the equation, and substitute x value into y = –x2 + x – 3.
Starting with x = -2
y = –x2 + x – 3
y = -(-2)2 + (-2) – 3
y = -(4) – 2 – 3
= -4 -2 – 3
= -9
Do for other x values:-
When x = - 1, y = -(-1)2 + (-1) – 3 = -5
When x = 0, y = -(0)2 + (0) – 3 = -3
When x = 1, y = -(1)2 + (1) – 3 = -3
When x = 2, y = -(2)2 + (2) – 3 = -5
When x = 3, y = -(3)2 + (3) – 3 = -9
Step 2: Fill in the values
X
|
-2
|
-1
|
0
|
1
|
2
|
3
|
Y
|
-9
|
-5
|
-3
|
-3
|
-5
|
-9
|
(b) Drawing the Graph
(b) Plot and draw the graph.
The graph is plotted by placing and joining the coordinates on the graph.
(c) Solving Equation using the Graph
There are two variables ( x and y) in the equation y = ax2 + bx + c.
<<a, b, c are constant (a specific number) >>
We can solve an equation when either x or y is given.
When y is given => find x,
When x is given => find y
When using a graph to solve an equation ax2 + bx + c,
Step 1 : Form an equation such that
y = ax2 + bx + c = k
=> y = k
Step 2 : Draw y = k on the graph
Step 3 : Solve the equation by locating the intersection of y = k and y = ax2 + bx + c and find the required value/s.
(c) Use the graph to solve the equation
(i) –x2 + x – 3 = -7
Step 1: Write and form the equation
(i) y = –x2 + x – 3 = -7
Step 2: ‘Equate’ both to get equation to plot/draw on graph
y = -7
Step 3: Draw the line and find the coordinate
Draw y = -7
From the graph, x = -1 2/5 and 2 2/5
(ii) –x2 + x – 3 = 0
y = –x2 + x – 3 = 0 (Step 1)
Y = 0 (Step 2)
Draw y = 0 (Step 3)
y = 3x2 – 2x + 4 = 12 (Step 1)
y = 12 (Step 2)
Draw the line and find coordinate(Step 3)
x = -4/3 or x = 2
(b) 3x2 – 2x – 3 = 0
Step 1: Write the equation
y = 3x2 – 2x + 4
y = 3x2 – 2x - 3 = 0 (Step 1)
Step 2: Equate
[need to ‘change’ equation to 3x2 – 2x + 4]
To equate, +7 needs to be "added"
y = 3x2 – 2x - 3 +7 = 0 + 7
y= 3x2 – 2x + 4 = 7
y = 7
Step 3: Draw the line y = 7 and find coordinate
x = -0.72 or x = 1.4
