highschool/Grade 9/Math/MCF3M1/Final_Exam_Study_Sheet.md

Study Sheet

Unit 1: Functions

Words to know:

• linear relation

• quadratic relation

• vertex of a parabola

• line of best fit

• axis of symmetry of a parabola

• intercepts

• Linear Relation: A relation which a single straight line can be drawn through every data point and the first differences are constant
  

• Non - Linear Relation: A single smooth curve can be drawn through every data point and the first differences are not constant

Relations

• A relation can be described using
  1. Table of Values (see below)
     
  2. Equations \(`(y = 3x + 5)`\)
     
  3. Graphs (Graphing the equation)
     
  4. Words
• When digging into the earth, the temperature rises according to the
• following linear equation: \(`t = 15 + 0.01 h`\). \(`t`\) is the increase in temperature in
• degrees and \(`h`\) is the depth in meters.

Definitions

• Parallel: 2 lines with the same slope
  
• Perpendicular: 2 lines with slopes that are the negative reciprocal to the other. They form a 90 degree angle where they meet.
  
• Domain: The ordered set of all possible values of the independent variable \(`x`\).
  
• Range: The ordered set of all possible values of the dependent variable \(`y`\).
  
• Continous Data: A data set that can be broken into smaller parts. This is represented by a Solid line.
  
• Discrete Data: A data set that cannot be broken into smaller parts. This is represented by a Dashed line.
  
• First Difference: the difference between 2 consecutive y values in a table of values which the difference between the x-values are constant.
  
• Collinear Points: points that line on the same straight line

Variables

• Independent Variable: A Variable in a relation which the values can be chosen or isn’t affected by anything.
  
• Dependent Varaible: A Variable in a relation which is dependent on the independent variable.

Scatterplot and Line of Best Fit

• A scatterplot graph is there to show the relation between two variables in a table of values.
  
• A line of best fit is a straight line that describes the relation between two variables.
  
• If you are drawing a line of best fit, try to use as many data points, have an equal amount of points onto and under the line of best fit, and keep it as a straight line.
  
• 

How To Determine the Equation Of a Line of Best Fit

1. Find two points ON the line of best fit
2. Determine the slope using the two points
3. Use point-slope form to find the equation of the line of best fit

Table of values

• To find first differences or any points on the line, you can use a table of values

• It shows the relationship between the x and y values.

• Use Finite differences to figure out if its quadraic or linear:

  • If the first difference is constant, then its linear. (degree of 1)
  • If the second difference is constant, then its quadratic. (degree of 2)

• This is a linear function

  x	y	First difference
  -3	5	\(`\cdots`\)
  -2	7	5-7 = 2
  -1	9	7-9 = 2
  0	11	9-11 = 2
  1	13	11-13 = 2
  2	15	15-13 =2

  • The difference between the first and second y values are the same as the difference between the third and fourth. The first difference is constant.

• This is a quadractic function

  x	y	First difference	Second difference
  5	9	\(`\cdots`\)	\(`\cdots`\)
  7	4	9-4 = 5	\(`\cdots`\)
  9	1	4-1 = 3	5-3 = 2
  11	0	1-0 = 1	3 - 1 = 2
  13	1	0-1 = -1	1 -(-1) = 2

  • The difference between the differences of the first and second y values are the same as the difference of the difference between the thrid and fourth. The second difference is constant.

Tips

• Label your graph correctly, the scales/scaling and always the independent variable on the x-axis and the dependent variable on y-axis
  
• Draw your Line of Best Fit correctly
  
• Read the word problems carefully, and make sure you understand it when graphing things
  
• Sometimes its better not to draw the shape, as it might cloud your judgement (personal exprience)
  
• Label your lines

Number of Solutions

• 

Discriminant

• The discriminant determines the number of solutions (roots) there are in a quadratic equation. \(`a, b , c`\) are the

• coefficients and constant of a quadratic equation: \(`y = ax^2 + bx + c`\)
  \(` D = b^2 - 4ac \begin{cases} \text{2 distinct real solutions}, & \text{if } D > 0 \\ \text{1 real solution}, & \text{if } D = 0 \\ \text{no real solutions}, & \text{if } D < 0 \end{cases} `\)

• 

Tips

• Read the questions carefully and model the system of equations correctly
  
• Be sure to name your equations
  
• Label your lines

Definitions

• Function: a relation which there is only one value of the dependent variable for each value of the independent variable (i.e, for every x-value, there is only one y-value).
• Vertical-line test: a test to determine whether the graph of a relation is a function. The relation is not a function if at least one vertical line drawn through the graph of the relation passes through two or more points.
• Real numbers: the set of real numbers is the set of all decimals - positive, negative and 0, terminating and non-terminating. This statement is expressed mathematically with the set notation \(`\{x \in \mathbb{R}\} `\)
• Degree: the degree of a polynomial with a single varible, say \(`x`\), is the value of the highest exponent of the variable. For example, for the polynomial \(`5x^3-4x^2+7x-8`\), the highest power or exponent is 3; the degree of the polynomial is 3.
• Function notation: \(`(x, y) = (x f(x))`\). \(`f(x)`\) is called function notation and represents the value of the dependent variable for a given value of the independent variable \(`x`\).
• Transformations: transformation are operations performed on functions to change the position or shape of the associated curves or lines.

Working with Function Notation

• Given an example of \(`f(x) = 2x^2+3x+5`\), to get \(`f(3)`\), we substitute the 3 as \(`x`\) into the function, so it now becomses \(`f(3) = 2(3)^2+3(3)+5`\).
• We can also represent new functions, the letter inside the brackets is simply a variable, we can change it.
  • Given the example \(`g(x) = 2x^2+3x+x`\), if we want \(`g(m)`\), we simply do \(`g(m) = 2m^2+3m+m`\).

Vertex Form

• Vertex from: \(`f(x) = a(x-h)^2 + k`\).
  • \(`(-h, k)`\) is the coordinates of the vertex

Axis of symmetry

• \(`x = -h`\)
• Example:
  • \(`f(x) = 2(x-3)^2+7`\)
  • \(`x = +3`\)
  • 

Direction of openning \(`\pm a`\)

• Given a quadratic in the from \(`f(x) = ax^2+bx+c`\), if \(`a > 0`\), the curve is a happy face, a smile. If \(`a < 0`\), the curve is a sad face, a sad frown.
• \(` \text{Opening} = \begin{cases} \text{if } a > 0, & \text{opens up} \\ \text{if } a < 0, & \text{opens down} \end{cases} `\)
• Examples
  • \(`f(x) = -5x^2`\) opens down, sad face.
  • \(`f(x) = 4(x-5)^2+7`\) opens up, happy face.

Vertical Translations \(`\pm k`\)

• \(` \text{Direction} = \begin{cases} \text{if } k > 0, & \text{UP }\uparrow \\ \text{if } k < 0, & \text{DOWN } \downarrow \end{cases} `\)

Horizontal Translations \(`\pm h`\)

• \(` \text{Direction} = \begin{cases} \text{if } -h > 0, & \text{shift to the right} \\ \text{if } -h < 0, & \text{shift to the left} \end{cases} `\)

• \(`f(x) = 1(x-4)^2`\)

  • \(`\uparrow`\) congruent to \(`f(x) = x^2`\)

  • Vertical Stretch/Compression

• \(`|a|\leftarrow`\): absolute bracket.

  • simplify and become positive

• \(` \text{Stretch/Compression} = \begin{cases} \text{if } |a| > 1, & \text{stretch by a factor of } a \\ \text{if } 0 < |a| < 1, & \text{compress by a factor of } a \end{cases} `\)

  • (Multiply all the y-values from \(`y = x^2`\) by a)
  • (Not congruent to \(`f(x) = x^2`\))

• Example of stretching

  • \(`f(x) = 2x^2`\) -Vertically stretch by a factor of 2

  • x	y
    -3	9(2) = 18
    -2	4(2) = 8
    -1	1(2)= 2
    0	0(2) = 0
    1	1(2) = 2
    2	4(2)= 8
    3	9(2) = 18

  • All y-values from \(`f(x) =x^2`\) are now multiplied by 2 to create \(`f(x)=2x^2`\)

• Example of compression

  • \(`f(x) = \frac{1}{2}x^2`\)
    • Verticallyc ompressed by a factor of \(`\frac{1}{2}`\)

  • x	y
    -3	9\(`(\frac{1}{2})`\) = 4.5
    -2	4\(`(\frac{1}{2})`\) = 2
    -1	1\(`(\frac{1}{2})`\) = \(`\frac{1}{2}`\)
    0	0\(`(\frac{1}{2})`\) = 0
    1	1\(`(\frac{1}{2})`\) = 1
    2	4\(`(\frac{1}{2})`\)= \(`\frac{1}{2}`\)
    3	9\(`(\frac{1}{2})`\) = 4.5

  • All y-values from \(`f(x) = x^2`\) are now multiplied by \(`\frac{1}{2}`\) to create \(`f(x) = \frac{x^2}{2}`\)