MCR3U Final Exam Review

Comprehensive practice flashcards covering Functions, Radicals, Quadratics, Trigonometry, Exponential Growth, Sequences, and Financial Math based on the MCR3U Final Exam Review.

Does the equation $x^2 = y + 10$ represent a function, and what is its domain and range?

Yes, it is a quadratic function because each $x$-value has only a single $y$-value. Domain: $\{x ∈ ℝ\}$, Range: $\{y ∈ ℝ | y ≥ -10\}$.

Why is the equation $x = y^2 - 9$ not considered a function?

It is not a function because the presence of $y^2$ means a single $x$-value may correspond to two different $y$-values.

What is the simplified form and restrictions for the expression $\frac{x + 5}{x - 3} - \frac{4x - 7}{2x + 1}$?

$\frac{-2x^2 + 30x - 16}{(x - 3)(2x + 1)}$ with restrictions $x \neq -\frac{1}{2}$ and $x \neq 3$.

Simplify the radical expression: $\sqrt{243} + 3\sqrt{12} - 8\sqrt{147}$.

$9\sqrt{3} + 6\sqrt{3} - 56\sqrt{3} = -41\sqrt{3}$.

In the bridge model $h = -0.043d^2 + 2.365d$, where $h$ is height and $d$ is horizontal distance (both in metres), what is the maximum height of the arch?

Approximately $32.52\,m$ (calculated by completing the square to find the vertex at $32.51875$).

What are the points of intersection for the system $y = -3x + 12$ and $y = 2x^2 - 11x + 16$?

$(2 + \sqrt{2}, -3\sqrt{2} + 6)$ and $(2 - \sqrt{2}, 3\sqrt{2} + 6)$. Scales to approximately $(3.41, 1.76)$ and $(0.59, 10.24)$.

State the transformations required to obtain $g(x) = -3\sqrt{\frac{1}{2}(x + 6)} + 4$ from the parent function $f(x) = \sqrt{x}$.

A vertical stretch by a factor of 3, a reflection in the $x$-axis, a horizontal stretch by a factor of 2, a translation 6 units left, and a translation 4 units up.

What is the inverse function $f^{-1}(x)$ for $f(x) = \frac{2}{4(x - 3)} + 1$ and what is its domain?

$f^{-1}(x) = \frac{2}{4(x - 1)} + 3$ with a domain of $\{x ∈ ℝ | x \neq 1\}$.

Find all angles $0^\circ \le \theta \le 360^\circ$ such that $\csc(\theta) = -\sqrt{2}$.

$\theta = 225^\circ$ and $\theta = 315^\circ$.

Identify two angles coterminal to $133^\circ$ within the range of $-360^\circ$ to $720^\circ$.

$493^\circ$ (by adding $360^\circ$) and $-227^\circ$ (by subtracting $360^\circ$).

In $\triangle DEF$ where $DE = 16\,m$, $EF = 14\,m$, and $\angle D = 56^\circ$, what are the possible values for $\angle F$?

$71^\circ$ or $109^\circ$ (due to the ambiguous case of the Sine Law).

Prove the trigonometric identity: $(1 - \cos^2(x))(1 + \frac{1}{\tan^2(x)}) = 1$.

$LS = (\sin^2(x))(\frac{\sin^2(x) + \cos^2(x)}{\sin^2(x)}) = \sin^2(x) (\frac{1}{\sin^2(x)}) = 1 = RS$.

State the amplitude, period, and phase shift for the function $y = 4\sin[3(x - 45^\circ)] + 1$.

Amplitude: $4$; Period: $120^\circ$; Phase shift: $45^\circ$ right; Vertical translation: $1$ unit up.

A mass on a spring hangs at rest $50\,cm$ above ground, is raised $40\,cm$ and released at $t = 0\,s$, oscillating sinusoidally every $1.5\,s$. What is the model equation?

$h(t) = 40\cos(240t) + 50$, where $k = \frac{360}{1.5} = 240$.

Simplify the expression and write as a fraction: $(\frac{16}{625})^{-\frac{3}{4}}$.

$\frac{125}{8}$.

A radioactive substance has an initial mass of $80\,mg$ and a half-life of $2.5$ days. How much remains after $8$ days?

$8.7\,mg$ (calculated using the equation $M(t) = 80(\frac{1}{2})^{t/2.5}$).

Determine the general term formula ($t_n$) for the arithmetic sequence $\frac{5}{8}, \frac{3}{4}, \frac{7}{8}, 1, \dots$.

$t_n = \frac{n}{8} + \frac{4}{8}$ or $t_n = \frac{1}{8}n + \frac{1}{2}$.

Determine the sum of the geometric sequence $729, 243, 81, \dots, \frac{1}{27}$.

$1093 \frac{13}{27}$ or $\frac{29524}{27}$, where $n = 10$.

Calculate the sum of the arithmetic series $-5, 1, 7, \dots, 133$.

$1536$, where $n = 24$.

Jamal wants to borrow $\$5000$ for $3$ years. Why is Option C ($6.2\%$ compounded weekly) better than Option A ($7\%$ simple interest) or Option B ($6.5\%$ compounded quarterly)?

Under Option C, he pays the least interest ($\$1021.44$) compared to Option A ($\$1050$) and Option B ($\$1067.04$).

How much should be deposited bi-weekly into an account earning $5.5\%$ interest compounded bi-weekly to save $\$500,000$ over $25$ years?

$\$358.62$ every two weeks.

How much must be invested at $6.4\%$ interest per year compounded monthly to pay a winner $\$1000$ every month for $25$ years?

$\$149,483.12$.