Graphs of Quadratic Functions

The expression x² + bx + c is called a quadratic function of x, where a, b and c are constants. When a quadratic function y = x² + bx + c is plotted, its graph gives a smooth curve called a parabola.

Consider \( \scriptsize y = x^2 + bx + c\)

Also, we can have Consider \( \scriptsize y = -ax^2 + bx + c\)

Example 9.1.1:

Plot the curve of y = x² + x – 6 for values of x from -4 to 3

(a) Use the curve to find :
(i) the values of x when y = 2.8
(ii) the value of y when x = 1.4
(iii) the minimum value of the function and the value of x

(b) Draw the axis of symmetry of the curve and write down its equation

Solution:

Plot the curve as shown below:

(a) From the graph:

(i) when y = 2.8, x = -3.5 or x = 2.5

(ii) when x = 1.4, y = -2.6

(iii) y_min = -6.25 at x = -0.5

(b) Equation axis of symmetry: x = \(– \frac {1}{2} \)

Scale:

• On x-axis 1 unit ≡ 1 cm
• On y-axis 1 unit ≡ 1 cm

Example 9.1.2:

Draw the curve of y = -x² for values of x from -4 to 4.

Use the curve to find:

(i) The maximum value of the curve
(ii) the equation of the line of symmetry
(iii) the value of x when y = -10
(iv) The value of y when x = 2.5

Solution:

From the curve:

(i) Maximum value = 0 i.e. at the origin we have the turning point.

(ii) The curve is symmetric about the y-axis thus the equation of the line of symmetry is x = 0

(iii) When y = -10, x = 3.2 or -3.2

(iv) When x = 2.5, y = -6.2

Scale:

• 1 unit = 1 cm on x-axis
• 2 units = 1 cm on y-axis

Example 9.1.3:

(a) Copy and complete the table of values of  y = 3 – 4x – 2x²

x	-4	-3	-2	-1	0	1	2
y

(b) Using a scale of 1 cm to 1 unit on the x-axis and 1 cm to 2 units on the y-axis, plot the graph of y = 3 – 4x – 2x² for values of x from -4 to 2

(c) Use the graph to find:

(i) the highest value of the function 3 – 4x – 2x²

(ii) the two values of x when y=-10

(iii)The coordinates of the point where y is 

(d) Draw the line of symmetry and state its equation

(e) For what range of values of x is y > 0?

Solution: (a)

x	-4	-3	-2	-1	0	1	2
y	-13	-3	3	5	3	-3	-13

(b)

(c)  (i) y_max = 5

(ii) x = -3.7 or x = 1.7

(d) Equation of line of symmetry: x = -1

(e) Range of x for y > 0;   -2.6 < x < 0.6

Scale:

• 1 unit ≡ 1 cm on x-axis
• 2 units ≡ 1 cm on y-axis

Example 9.1.4:

(a) Draw the graph of y = 2x² -3x – 5 for -3 ≤ x ≤ 4 and hence use the graph to solve the equation 2x² – 3x – 5 = 0

(b) Use your graph to solve the equation 2x² + x – 2 = 4x + 8

Solution:

Scale:

• 1 unit ≡ 1 cm on x-axis
• 5 units ≡ 1 cm on y-axis

(a) The roots of the equation at a point where the graph cuts the x-axis i.e. x = -1 or x = 2.5

(b) 2x² + x -2 = 4x + 8

Subtract 4x from both sides 

⇒ 2x² – 3x – 2 = 8

subtract 3 from both sides

⇒ 2x² – 3x – 5 = 5

⇒ y = 5

From the graph x = -1.5 or x = 3.1

Scale:

• 1 unit ≡ 1 cm on x-axis
• 5 units ≡ 1 cm on y-axis