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MCF3M UNIT 1 – QUADRATIC FUNCTIONS

PREREQUISITE SKILLS

Determine the value of y when x = -2 in the following.

a) y = 3x – 7b) y = 5x2 – 3x + 2c) 2x – 4y = 16d) x2 + y2 = 13

Complete the following table of values

a) y = -2x + 5 b) y = x2 - 6x + 9

Find the image of the point (4, -5) after each transformation

a)Translation 3 units to the left

b)Translation of 6 units up

c)Reflection in the x-axis

Determine the surface area of the following Determine the volume of the box

4cm

6cm 22cm4.2cm

2.1cm

6.8cm

18 cm

MCF3M1.1 IDENTIFY FUNCTIONS

A FUNCTION is a set of ordered pairs in which no two ordered pairs have the same x-coordinate. This means that each x can only produce 1 possible y.

A function can be represented in different ways. The requirement that each ordered pair must have different x-coordinate can be seen by using:

a) a table of valuesb) a graphc) an equationd) a mapping diagram

ex. A linear function can be written as

x y

x y y = 3x -1

-2 -7-2 -7

-1 -4-1 -4

0 -1 0 -1

1 2 -1 1 2

2 5 2 5

Tests for a FUNCTION:

  • Equation Test for a Function:

 If a value of x can be found that produces more than one value of y, the equation does not represent a function.

Example: Determine if the following equations represent a function.

a) y = x2b) y2 = x or y = 

  • Vertical Line Test:

When a graph is given, visualize a vertical line moving across the graph.

If the vertical line intersects the graph in more than one point, the graph does not represent a function.

Example: Determine if the following graphs represent a function.

Mapping Diagrams:

If a mapping diagram represents a function, it is not possible to have two or more arrows starting at the same number in the x column.

Ex. Write each mapping diagram as a set of ordered pairs and determine whether the following mapping diagrams represent functions:

x yxy

a) b)

0 -3 -3 5

-2 -2

1 -1 -1 0

0 0

4 1 1 -3

2 2

9 3 3 -4

Ex. Graph y = x2 as a mapping diagram using x values between -3 and 3 inclusive.

To represent functions, we use notations such as f(x) and g(x).

Ex. Linear function: y = 2x + 1 In function notation : f(x) = 2x + 1

  • The notation f(x) is read “f of x” or “f at x”. It means that the expression that follows contains x as a variable.

For example: f(3) means substitute 3 for every x in the expression and solve for y, or f(3).

ex. Quadratic function : f(x) = x2 – 4x + 7

f(3) = (3)2 – 4(3) + 7

= 9 – 12 + 7

= 4

 when x = 3, y = 4 or  f(3) = 4

Example: Given f(x) = x2 – 5, determine each value.

a) f(2)b) f

Example: For the graph of y = f(x) below,

a)Determine f(0) and f(-3),

b)Solve the equation f(x) = -2.

MCF3M1.2 DOMAIN AND RANGE

DOMAIN AND RANGE of a Function

  • The domain of a function is the set of all x-values (input numbers) of the function.
  • The range of a function is the set of all y-values (output numbers) of the function.

range

domain

Examples :

  1. Given the following relations, state the domain and range of each function.

a) b) c)

f)

d) {(1, 2), (3, 4), (4, 6), (7, 10)} e)

  1. A ball is thrown from the top of a 14m building. The ball is thrown up into the air 2.5m and falls to the ground after 3.2seconds.

a)Sketch a graph the path of the ball.

b)Write the domain of the relation.

c)Write the range of the relation.

  1. Consider these situations and determine the domain and range.

a)A gas station sold between 30 and 60 litres of gas at $1.00 per litre.

b) A variety store sold between 30 and 50 newspapers at $0.75 each.

  1. Given the equations of the following functions, state their domain and range.

a) y = x – 5 b) y = x2

MCF3M 1.3 ANALYSE QUADRATIC FUNCTIONS

QUADRATIC FUNCTION

  • A function of the form f(x) = ax2 + bx + c where a, b and c are real numbers and a ≠ 0.

The graph of a quadratic FUNCTION is called a PARABOLA.

The VERTEX of a parabola is the point with the greatest y coordinate (if the graph opens down) or the smallest y coordinate (if the graph opens up).

·If the graph opens down, the vertex is the MAXIMUM point.

·If the graph opens up, the vertex is the MINIMUM point.

The DIRECTION OF OPENING can be determined by the sign of the second differences column.

·If the constant value of the 2nd differences is positive, the graph opens up.

·If the constant value of the 2nd differences is negative, the graph opens down.

The AXIS OF SYMMETRY is a vertical line that goes through the vertex of a parabola.

·The equation of the axis of symmetry is x = the x coordinate of the vertex.

·The axis of symmetry is the perpendicular bisector of the line segment joining any two points on the parabola that have the same y-value.

·

1

Example: Determine whether the following relations are linear or quadratic. If they are quadratic, determine the direction of opening of the parabola.

a) b)

Drawing the BASE curve of a parabola y = x2

x / y
-3
-2
-1
0
1
2
3

MCF3M1.4 STRETCHES OF FUNCTIONS

  • When we transform the graph of y = x2 to y = ax 2, the following transformations occur:
  • If a is negative, the parabola is reflected in the x-axis.

  • If a  1 or a  -1, the parabola is vertically stretched by a FACTOR OF a.

  • If –1  a  1, the parabola is vertically compressed by a FACTOR OF a, or we can say the parabola is horizontally compressed by a factor of a.

To graph a vertical stretch by a factor of ‘a’

  • Multiply each corresponding y-coordinate by ‘a’.

To graph a reflection in the x-axis

  • Multiply each corresponding y-coordinate by ‘-1’.

Examples: For each of the following functions describe the transformation(s) in words and then graph the original y = x2 and the transformed function on the same grid.

a) y = 2x2

b) y = -0.5x2

Example: Determine the stretch factor if the point (1, -2) is on the graph of the function y = ax2.

MCF3M1.5 TRANSLATIONS OF FUNCTIONS

A TRANSLATION is a slide or shift; a transformation that moves a graph right or left, up or down. The shape and size of the graph is not changed by a translation.

  • When we transform the graph of y = x2 to y = (x – h)2 + k, the following transformations occur:

h  If h is positive, the parabola is translated horizontallyh units to the RIGHT.

 If h is negative, the parabola is translated horizontallyh units to the LEFT.

k  If k is positive, the parabola is vertically translated UP by k units.

 If k is negative, the parabola is vertically translated DOWN by k units.

Examples: For each of the following, state the transformation(s) relative to the graph of y= x2, write the coordinates of the vertex and sketch the graph.

a)y = (x + 6)2

b) y = x2 + 2

c) y = (x – 2)2 – 3

Example: The x-intercepts of a parabola that opens upward are -3 and 5. What can we determine from the x-intercepts of the parabola? Determine the equation of the parabola in the form f(x) = (x – k)2 + k.

MCF3M1.6 SKETCH GRAPHS USING TRANSFORMATIONS

RECAP:

  • When we transform the graph of y = x2 to y = a(x – h)2 + k, the following transformations occur:

a  If a is negative, the parabola is reflected in the x-axis.

If a  1 or a  -1, the parabola is vertically stretched by a FACTOR OF a.

If –1  a  1, the parabola is vertically compressed by a FACTOR OF a, or we can say the parabola is horizontally compressed by a factor of a.

h  If h is positive, the parabola is translated horizontallyh units to the RIGHT.

 If h is negative, the parabola is translated horizontallyh units to the LEFT.

k  If k is positive, the parabola is vertically translated UP by k units.

 If k is negative, the parabola is vertically translated DOWN by k units.

Examples: For the function f(x) = 2(x + 3)2 – 5,

a)Describe the transformations relative to the graph of y = x2.

b)Write the coordinates of the vertex and the equation of the axis of symmetry.

c)Sketch the graph and the graph of y = x2.

d)State the domain and range.

Tips for graphing:

  • Do the vertical stretch/compression first.
  • Do the reflection second.
  • Move the points right/left, up/down last.

Example: Graph y = -2(x + 7)2 - 4 and the original y = x2 on the graph below.

Examples: Write the equation for a parabola that satisfies each set of conditions.

a) Vertex at (-2, 0); opens down; same shape as y =3x2.

b)Vertex at (2, 6); y-intercept -1.

c)The graph of y = x2 is stretched vertically by a factor of 5/2, then translated down 7.

d)The graph of y = x2 is reflected about the x-axis, stretched vertically by a factor of 3, then translated right 6 units and up 5 units.

e)Opens downward, congruent in shape to y = 2x2 and has x-intercepts of 4, -2.