Graphs of Quadratic Functions

Worked example 12

(Sketching the quadratic graph in the form of y=(x-m)(x-n)

Sketch the graph of each of the following functions:

a) y=(x-1)(x+3)

b) y=-(x+1)(x-5)

Solutions for a)

Step 1: Determine whether the coefficient of x² is negative or positive.

Since the coefficient of x² is 1 which is a positive number, the graph opens upwards.

Step 2: Obtaining the x-intercepts by substituting y=0 into the equation.

When y=0,

(x-1)(x+3)=0

(x-1)=0 or (x+3)=0

x=1 x=-3

∴ the x intercepts are 1 and -3

Step 3: Obtaining the y -intercepts by substituting x=0 into the equation.

When x=0,

y=(0-1)(0+3)

y=-3

∴ the y intercept is -3

Step 4: Complete the graph.

Solutions for b)

Step 1: Determine whether the coefficient or x² is negative or positive

Since the coefficient of x² is =1 which is a negative number, the graph opens downwards.

Step 2: Obtaining the x -intercepts by substituting y=0 into the equation.

When y=0,

-(x+1)(x-5)=0

(x+1)=0 or (x-5)=0

x=-1 x=5

∴ the x intercepts are -1 and 5

Step 3: Obtaining the y-intercepts by substituting x=0 into the equation

When x=0,

y=-(0+1)(0-5)

y=5

∴the y intercept is 5

Step 4: Sketch the graph.

Practice NOW 12

Sketch the graph of each of the following functions.

a) y=(x-2)(x-6)

b) y=-(x-3)(x+2)

c) y=(1-x)(x+3)

WORKED EXAMPLE 13

Given the quadratic function y=-(x-2)+9²,

(i) find the coordinates of the x- and y-intercepts,

(ii) write down the coordinates of the maximum point of the graph,

(iii) sketch the graph,

(iv) state the equation of the line of symmetry of the graph.

Solutions for (i)

When y=0,

0=-(x-2)²+9 Solve x by making it the subject

(x-2)² = 9

(x-2)=±9

x=3+2 or x=-3+2

x=5 or x=-1

∴ the coordinates of x-intercepts are (5,0) and (-1,0).

When x=0,

y = -(0-2)²+9

y=5

∴ the coordinates of y-intercept are (0,5).

Solution for (ii)

The coordinates of the maximum point are (2,9)

Solutions for (iii)

Step 1: Determine whether the coefficient of x² is negative or positive.

Since the coefficient of x² is -1 which is a negative number, the graph opens downwards.

Step 2: Obtaining the x-intercepts by substituting y=0 into the equation.

From (i) the x-intercepts are 5 and -1

Step 3: Obtaining the y-intercepts by substituting x=0 into the equation.

From (i) the y intercept is 5

Step 4: Sketch the graph and include the maximum point.

Solution for (iv)

The equation of the line of symmetry is x=2.

PRACTICE NOW 13

1. Given the quadratic function y=-x(-3)²+4,

(i) find the coordinates of the x- and y-intercepts,

(ii) write down the coordinates of the maximum point of the graph,

(iii) sketch the graph,

(iv) state the equation of the line of symmetry of the graph.

 

2. Given the quadratic function y=(x+2)²-16,

(i) find the coordinates of the x- and y-intercepts,

(ii) write down the coordinates of the maximum point of the graph,

(iii) sketch the graph,

(iv) state the equation of the line of symmetry of the graph.

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