Graphs of Quadratic function

1. 15.3

2. x² ?x
?x +8
x
x
+2
+4
Add
to
+6x
Pairs of numbers
that multiply to
make “+8”:
+1 and +8
-1 and -8
+2 and +4
-2 and -4
Which pair adds to
make “+6”?

3. x² ?x
?x +16
x
x
-2
-8
Add
to -
10x
Pairs of numbers
that multiply to
make “+16”:
+1 and +16
-1 and -16
+2 and +8
-2 and -8
+4 and +4
-4 and -4
Which pair adds to
make “-10”?

4. By factorising, solve the equations
1) x
2
+ 9x + 20 = 0 2) x
2
– x – 2 = 0
3) x
2
+ 4x – 32 = 0 4) x
2
– 7x + 10 = 0
5) x
2
+ 3x – 4 = 0 6) x
2
+ 10x + 25 = 0
7) x
2
+ 2x – 3 = 0 8) x
2
– 8x + 16 = 0
9) x
2
+ 10x + 9 = 0 10) 2x
2
+ 5x – 12 = 0
x = - 4 or - 5 x = - 1 or 2
x = - 8 or 4 x = 2 or 5
x = - 4 or 1 x = - 5
x = - 3 or 1 x = 4
x = 1.5 or -4
x = - 9 or - 1

5. Example:
Write 𝑥2 + 8𝑥 − 3 in the form (𝑥 + 𝑎)2+𝑏.
Start by doing this: (𝑥 + 4)2−16 − 3
Half the 8
Square the
4
From the
question
Answer: (𝒙 + 𝟒)𝟐
−𝟏𝟗

6. Example:
Write 𝑥2 − 10𝑥 + 7 in the form (𝑥 + 𝑎)2+𝑏.
Start by doing this: (𝑥 − 5)2−25 + 7
Half the -10
Square the
5
From the
question
Answer: (𝒙 − 𝟓)𝟐
−𝟏𝟖

7.  Complete the square
1) x
2
+ 12x + 5 2) x
2
– 4x – 11
3) x
2
+ 8x – 13 4) x
2
– 16x + 22
5) x
2
+ 5x – 3
(x + 6)2 - 31 (x - 2)2 - 15
(x + 4)2 - 29 (x - 8)2 - 42
(x + 2.5)2 – 9.25

8.  We can tell a lot about a graph without even
plotting it – just by using the equation (name)
1. The Roots (where it crosses the x-axis)
2. The Intercept (where it crosses the y-axis)
3. The Turning Point (where the gradient
changes from negative to positive or vice
versa)
4. Whether the turning point is a Maximum
point or Minimum point

9. x
2
+ 2x - 3 = 0
Factorised:
(x + 3) (x - 1) = 0
Therefore:
x = -3 and x = 1
Look at the graph

10. The Roots of a quadratic are where the graph
intersects with the x-axis.
They can be found by solving the quadratic
equation when y = 0.
This can be done by factorising, completing
the square or using the quadratic formula.

11. y = x
2
+ 2x - 3
We can also see the
intercept

12. The Intercept is where the graph cuts the y-
axis.
This is always the (+c) (end number) of the
quadratic equation

13. y = x
2
+ 2x – 3
Completed Square:
y = (x + 1)2 – 4

14. The turning point of a quadratic is where the
gradient changes from negative to positive or
vice versa.
This can be found by completing the square in
the form y = a(x + b)2 + c
The coordinate of the turning point is (-b, c)

15. y = x
2
+ 2x – 3
If a quadratic is
positive (U Shape)
the turning point will
be a Minimum.
If it is a negative (n
Shape) it will be a
maximum

16. The Turning Point is either a Maximum value or
a Minimum value.
If a quadratic is positive (U Shape) the turning
point will be a Minimum.
If it is a negative (n Shape) it will be a
Maximum

17. Using these 4 attributes
of a quadratic function
we can sketch the
graph.
For example:
y = x2 + 10x + 9
1. Roots:
(x+9) (x+1) = 0
x = -9 x = -1
(-9, 0) (-1, 0)
2. Intercept:
+9 (0, 9)
3. Turning Point:
y = (x+5)2 -16
(-5, -16)
4. Max/Min:
Minimum

18. Lets plot these
y = x2 + 10x + 9
1. Roots:
(-9, 0) (-1, 0)
2. Intercept:
(0, 9)
3. Turning Point:
(-5, -16)
4. Max/Min:
Min

19. y = x2 + 6x + 8
1. Roots: (-4, 0) (-2, 0)
2. Intercept: (0, 8)
3. Turning Point: (-3, -1)
4. Max/Min: Min