Introduction to Quadratic Curves

THE QUADRATIC CURVE

Parabolic Path of the Football is a Quadratic Curve.

THE QUADRATIC EQUATION

A quadratic equation is one in which 2 is the
highest of power of the variable(s) (or
unknown(s)) in the equation.

The general form of a quadratic equation in
x is ax²+ bx + c, where a, b and c are
constants.

THE QUADRATIC FUNCTION

The general form is

y = ax²+ bx + c …equation 1

Now consider y = 1x²+ 3x –1…equation 2

Comparing equations 1 & 2
a = 1 the coefficient of x²

b = +3 the coefficient of x

c = -1 the y-intercept

Multiple Choice

In the equation
y = x² + 5x + 7, match each leading coefficient with its correct letter 

• a=0, b=5, c=7
• a=1, b=5, c=7
• a=7, b=5, c=1

Multiple Choice

Identify b of the following quadratic function:
 y = 2x ² − 4x − 2

• 2
• 4
• -4
• -2

KEY COMPONENTS OF A QUADRATIC CURVE (PARABOLA)

QUADRATIC CURVE (PARABOLA)
CASE 1: COEFFICIENT OF X²>0

THE QUADRATIC

EQUATION

y = x²+ 3x – 1

a =1 that is a > 0 so the curve
opens up(notice the U –shape)

Note the y-intercept is the point
(0, -1), that is when x = 0.

QUADRATIC CURVE (PARABOLA)
CASE 2: COEFFICIENT OF X²< 0

THE QUADRATIC

EQUATION

y = 3 + 2x - x²

Note the y-intercept is the point (0, 3), that is when x = 0.

 The maximum point is (1,4)

 The roots occur when y = 0
that is at points (-1,0) &
(3,0)

THE QUADRATIC

EQUATION

y = x² - 3x + 2

By factorising

y = (x - 1)(x - 2)

For roots, y = 0,this implies

(x - 1)(x - 2) = 0 so

either x-1= 0 or x-2 = 0

Then x = 1 or x = 2

Therefore the coordinates of
the roots are (1,0) & (2,0).

when x = 0, that is, where the
curve cuts the y-axis.

When x = 0 ,

y = 0² - 3(0) + 2 = 2

The coordinates of the y-
intercept are (0,2).

THE QUADRATIC

EQUATION
y = x² - 3x + 2

a = 1 i.e. a > 0, the
curve opens up and
has a minimum point.
The minimum point is
(1.5, -0.25).

The line x = 1.5 divides
the curve in two
halves. It is also called
the axis of symmetry.

Multiple Choice

Which of the following quadratic functions opens down?

• y = x² - 6x + 3

• y = -4x² + 9

• y = 0.5x² - 9x + 6

• y = 16x² + 14x + 9

Multiple Choice

If a parabola opens up, then it has a maximum.

• True

• False

FINDING THE X & Y-INTERCEPTS OF A QUADRATIC CURVE

Multiple Choice

What are the coordinates of the y-intercept?

• (-3, 0)
• (0, -3)
• (2, 1)
• y=-3

Multiple Choice

What is the y-intercept of 
f(x) = 2x² + 4x + 6

• (6 , 0)
• (-6 , 0)
• (0, -6)
• (0 , 6)

Multiple Choice

The equation y = x² + 3x - 18 is graphed.  Based on this graph, what are the roots (x-intercepts) of the equation x² + 3x - 18 = 0?

• -3 and 6
• 0 and -18
• 3 and -6
• 3 and -18

EXPANDING BRACKETS & FACTORISING

Watch the following videos and make notes.

Re-watch the videos, if necessary,
ensure that you spend enough time reviewing.

VIDEO 1: EXPANDING BRACKETS

​https://youtu.be/63oU-AIzTcE?si=jEDsMdBIUVEnDNcg

VIDEO 2: FACTORISING QUADRATICS

VIDEO 3: FACTORISING QUADRATICS

VIDEO 4: FACTORISING QUADRATICS

Multiple Choice

What are the x-intercepts of the graph of y = -x² - 6x + 40?

(HINT: Factor out a negative from the equation)

• -11 and 5

• -7 and 1

• -10 and 4

• 4 and 10

Multiple Choice

Which of the following is a solution of y = -3x² + 22x + 9?

• -3

• 31/6

• 3

• -7

Multiple Choice

Which one of the following is a root of y = 4x² - 17x + 13?

• 1

• 13

• -17 / 3

• -1

​Axis of Symmetry:
x = h --> x = 1

​h =

​k =

Multiple Choice

Which of the following is the equation for the axis of symmetry?

• x = b / 2a

• x = -a / 2b

• x = -b / 2a

• x = -c / 2a

Multiple Choice

Write the equation for the axis of symmetry for the quadratic function y = 3x² + 8x - 6.

• x = 4 / 3

• x = -4 / 3

• x = -3 / 4

• x = -2

Multiple Choice

Find the coordinates of the vertex of the quadratic function y = 4x² + 8x - 3.

• (1, -7)

• (-1, -7)

• (-2, 45)

• (-1, 7)

Multiple Choice

What is the vertex of the quadratic function:
y = x² - 6x + 11?

• (3, 2)

• (-3, -2)

• (-3, 2)

• (3, -2)

Multiple Choice

Which quadratic has a vertex at $$\left(2,6\right)?$$

• $$y=x^2-4x+10$$

• $$y=2x^2-8x+6$$

• $$y=-2x^2-8x+2$$

• $$y=-x^2+2x+5$$

Re-watch the videos, if necessary, ensure that you spend enough time reviewing.

Practice doesn't make perfect, it makes permanent!!!