Foundations of Mathematics 11: Study Guide
Unit 4: Quadratic Functions
1. Algebraic Forms of a Parabola
A quadratic function (degree 2) creates a U-shaped graph called a parabola. Depending on the
information given, we use one of three forms:
●
●
●
Standard (Vertex) Form:
○ The vertex is exactly
.
○ The axis of symmetry is the vertical line
Factored Form:
○ The x-intercepts are and .
.
○ The axis of symmetry is halfway between them:
General Form:
○ The y-intercept is always the constant .
.
2. Graphical Properties
●
●
Direction of Opening: If is positive, the parabola opens up (has a minimum); if
negative, it opens down (has a maximum).
Domain and Range: The domain is always all real numbers (
). The range
depends on the vertex and the opening direction.
○ Example: If the vertex is
and it opens down, the range is
is
.
3. Desmos Procedures
1. Finding Key Points: Type the equation. Click on the curve to see grey dots. The
top/bottom dot is the vertex. Dots on the x-axis are roots/zeros. The dot on the y-axis is
the y-intercept.
2. Projectile Motion: When modeling a path like
, the vertex yvalue is the maximum height, and the x-intercept is the time it hits the ground.
Extra Practice:
●
●
Q1: A flare follows
.
○ What is the maximum height? __________
○ What was the height at the start (
)? __________
Q2: Convert
to General Form.
Unit 5: Systems of Equations & Inequalities
1. Solving Systems
●
●
Solutions: The solution to a system is the point of intersection (
) where two
graphs cross.
Nonlinear Systems: A system can involve a line and a parabola or two parabolas,
resulting in 0, 1, or 2 solutions.
2. Linear Inequalities in Two Variables
To graph an inequality like
●
●
:
Boundary Line: Use a solid line for or (includes the line). Use a broken line for
or (excludes the line).
Shading: Shade the region of "possible" answers. In Desmos, this happens
automatically.
3. Linear Programming (Optimization)
Used to find the "best" outcome (e.g., maximum profit) under constraints.
1.
2.
3.
4.
5.
Define Variables: Let = item A, = item B.
Constraints: Write inequalities (e.g., "at most 20 hours" becomes
).
Feasible Region: The area where all shaded regions overlap.
Objective Function: The equation you want to maximize, like
Vertex Test: The optimal solution is always at one of the vertices (corners) of the
feasible region.
Extra Practice:
●
Q3: Find the intersection of
●
Q4: A region has vertices at
?
and
.
and
. Which point maximizes
.
Unit 6: Statistics
1. Central Tendency and Dispersion
●
●
Mean ( ): The average. Adding a constant to data increases the mean by that constant.
Standard Deviation ( ): Measures consistency. A low means data is close to the
average; a high means it is spread out. Adding a constant does not change .
2. The Normal Distribution
Data often forms a symmetrical "Bell Curve".
●
68-95-99.7 Rule: 68.3% of data is within
●
z-Scores: The number of standard deviations a value is from the mean.
; 95.4% within
; 99.7% within
3. Confidence Intervals
Used when estimating a population from a sample.
●
Example: "42% support,
margin of error, 19 times out of 20."
○ Confidence Level: 95% (from 19/20).
○ Confidence Interval: 39% to 45%.
○ Sample Size Rule: Larger samples lead to smaller margins of error (more
precision).
Extra Practice:
●
Q5: Marks have
●
Q6: If
and
and
. Find the z-score for a mark of 81.
, what % of data is between 90 and 110?
.
Unit 7: Financial Mathematics
1. Simple vs. Compound Interest
●
●
Simple Interest (
Compound Interest (
Growth is exponential.
●
Rule of 72: Years to double
): Earned only on the original amount. Growth is linear.
): Earned on principal and accumulated interest.
.
2. TVM Solver (Online Tool)
For all complex calculations, use these inputs:
●
●
●
●
●
●
N: Total payments (Years P/Y).
I%: Annual rate (e.g., 5 for 5%).
PV: Present Value (Amount started/borrowed).
PMT: Periodic payment.
FV: Future Value (Target amount or 0 if paid off).
Sign Convention: Money leaving you is negative (investments/payments); money
coming to you is positive (loans/withdrawals).
3. Mortgages and Loans
●
●
●
Finance Charge: The total interest paid (
).
Canadian Mortgages: Interest is compounded semi-annually (
).
Accelerated Weekly Payments: Dividing a monthly payment by 4 and paying weekly
leads to 13 months of payments per year, paying the loan off much faster.
Extra Practice:
●
●
Q7: How much simple interest is earned on $1,200 at 5% for 9 months (
)?
Q8: A car costs $10,000. You pay $200/month for 5 years. What is the finance charge?
Technology Checklist
●
●
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Desmos: Used for Units 4 and 5 to find vertices, intercepts, and feasible region corners.
TVM Solver: Used for Unit 7 to calculate monthly payments and savings goals.
Scientific Calculator: Used for Unit 6 to calculate z-scores and basic averages.