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

Graphs of Quadratic Functions

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

Screen Shot 2021 01 27 at 5.16.27 PM

Hint: The highest power of x is 2 i.e. a second degree function

  • The points T1 and T2 are the turning points i.e. where the curve changes direction (inflexion point)
  • When a is positive, we have a minimum curve with shape and the minimum value of the function ymin is at the turning point T1 and the value of x where it occurs gives the equation of the line of symmetry.
  • When a is negative, we have a maximum curve shape and the maximum value of the function ymax is at the turning point T2 and the value of x where it occurs gives the equation of the line of symmetry.
  • The line of symmetry or the axis of symmetry divides the curve into two equal parts.

Example 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:

X-4-3-2-10123
X2169410149
X-4-3-2-10123
-6-6-6-6-6-6-6-6-6
Y60-4-6-6-406
Screen Shot 2021 01 27 at 5.44.45 PM

(a) From the graph:

(i) when y = 2.8, x= -3.5 or x = -2.6

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

(iii) ymin. = -6.4 at x = -0.5

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

Scale: On x-axis 1unit ≡1cm

           On y-axis 1unit ≡ 1cm

Example 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

X-4-3-2-101234
Y-16-9-4-10-1-4-9-16

image4
Quadratic equation Graph for y=-x2

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: 1unit =1cm on x-axis

2units = 1cm on y-axis

Example 3

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

X-4-3-2-1012
Y

(b) Using a scale of 1cm to 1unit on the x-axis and 1cm to 2units 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-4-3-2-1012
Y-13-3353-3-13

(b)

image5
Diagram on Quadratic Equation

(c)    (i) ymax = 5

(ii) x = -3.7 or x = 1.7

(iii) Coordinate of ymax = (-1.5)

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

(e) Range of x for y > 0;   -2.5< x < 0.5

Scale: 1unit ≡ 1cm on x-axis | 2units ≡ 1cm on y-axis

Example 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:

X-3-2-101234
2x218820281832
-3x9630-3-6-9-12
-5-5-5-5-5-5-5-5-5
y2290-5-6-3415

Scale: 1unit ≡ 1cm on x-axis.     5units ≡ 1cm on y-axis

image6
Diagram on Quadratic Equation

(a) The roots of the equation at 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 

i.e.  2x2 – 3x – 2 = 8

subtract 3 from both sides

i.e. 2x2 – 3x – 5 = 5

i.e. y = 5

From the graph x = -1.6 or x = 3.2

Scale: 1unit ≡ 1cm on x-axis and 5units ≡ 1cm on y-axis

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