MHF4U Final Exam Review: Free Practice Exam

Free MHF4U Practice Exam

21 questions, up to 3 from each of the 7 units, taken from the free tests on Martian Lab. Do them in one sitting, on paper, then open the answers.

1. Polynomial Functions

   Distinguish, using words, the difference between an odd degree function, an odd order zero, and an odd function. Include examples in your explanation.

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   - odd degree function can be any polynomial function with an odd degree. - an odd function must be a odd degree function that has a point at the origin and the graph is rotated $180^\circ$ about the origin - an odd order states the way the graph crosses the $x$-axis (order 1 is straight through, order 3 is a twist)

2. Polynomial Functions

   The given table of values represents a polynomial function. Determine an equation for the function.

   

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   $k = a \cdot n!$ $-6 = a \cdot 3!$ $-1 = a$ $y = -x^3 + bx^2 + cx + 6$ plug (1, 8): $8 = -(1)^3 + b(1)^2 + c(1) + 6$ $8 = -1 + b + c + 6$ $3 = b + c \quad \cdots (1)$ plug (2, 0): $0 = -(2)^3 + b(2)^2 + c(2) + 6$ $0 = -8 + 4b + 2c + 6$ $2 = 4b + 2c \quad \cdots (2)$ (2) - (1): $2 = 4b + 2c$ $-3 = b + c$ $-1 = 3b + c$ $-1 - 3b = c \quad \cdots (3)$ sub (3) into (1): $3 = b + (-1 - 3b)$ $4 = -2b$ $-2 = b$ sub b into (2): $2 = 4(-2) + 2c$ $10 = 2c$ $5 = c$ $\therefore$ the equation is $y = -x^3 - 2x^2 + 5x + 6$

3. Polynomial Functions

   Determine whether $f(x) = -x(x-3)^2 + 3x(2x-3)$ is even, odd, or neither. Justify algebraically.

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   $f(-x) = -(-x)[(-x)-3]^2 + 3(-x)(2(-x)-3)$ $= x(-(x+3))^2 - 3x(-2x-3)$ $= x(x+3)^2 - 3x(-2x-3)$ $f(-x) \neq f(x)$ $\therefore$ not even $-f(x) = -[-x(x-3)^2 + 3x(2x-3)]$ $= x(x-3)^2 - 3x(2x-3)$ $f(-x) \neq -f(x)$ $\therefore$ not odd $\therefore$ the function is neither

4. Polynomial Equations

   Divide $8x^{3} - 10x^{2} + x - 5$ by $2x - 3$ then relate $8x^{3} - 10x^{2} + x - 5$ and $2x - 3$ in two different ways (i.e. using a division statement and using quotient form). The division statement is: $\rule{0.5cm}{0.4pt}$ The quotient form is: $\rule{0.5cm}{0.4pt}$

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   See full solution in image below. The division statement is: $8x^{3} - 10x^{2} + x - 5=(4x^{2}+x+2)(2x-3)+1$ The quotient form is: $\dfrac{8x^{3} - 10x^{2} + x - 5}{2x - 3}=4x^{2}+x+2+\dfrac{1}{2x-3}$

5. Polynomial Equations

   Solve $3x^{4} + x^{3} - 12x^{2} - 4x < 0$ for $x \in R$, algebraically. Show your solution using a Real Number Line.

   

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   $x(3x^{3}+x^{2}-12x-4)<0$ $x(x^{2}(3x+1)-4(3x+1))<0$ $x(x^{2}-4)(3x+1)<0$ $x(x-2)(x+2)(3x+1)<0$ $x=0,\;2,\;-2,\;-\dfrac{1}{3}$ $\therefore \; f(x) < 0$ when $x\in\left(-2,-\dfrac{1}{3}\right)\cup(0,2)$

6. Polynomial Equations

   Find the equation of the quartic function in STANDARD FORM, with zeros $\pm 1, 2 \pm \sqrt{3}$ and passes through the point $(2, 5)$.

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   $f(x)=a(x-1)(x+1)\left[x-(2+\sqrt{3})\right]\left[x-(2-\sqrt{3})\right]$ $=a(x^{2}-1)(x-2-\sqrt{3})(x-2+\sqrt{3})$ $=a(x^{2}-1)\left[(x-2)-\sqrt{3}\right]\left[(x-2)+\sqrt{3}\right]$ $=a(x^{2}-1)\left[(x-2)^{2}-3\right]$ $=a(x^{2}-1)(x^{2}-4x+1)$ $=a(x^{4}-4x^{3}+4x-1)$ $5=a(16-32+8-1)$ $a=-\dfrac{5}{9}$ $f(x)=-\dfrac{5}{9}x^{4}+\dfrac{20}{9}x^{3}-\dfrac{20}{9}x+\dfrac{5}{9}$

7. Rational Functions

   What is the domain of the function $f(x) = \dfrac{12}{5x-1}$?

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   $\left\{x \in R,\ x \neq \dfrac{1}{5}\right\}$

8. Rational Functions

   What is the horizontal asymptote of $f(x) = \dfrac{6-2x}{6x-3}$?

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   $y = \dfrac{-1}{3}$

9. Rational Functions

   What are the zeros of $f(x) = \dfrac{(x^2-4)(x-1)}{(x-2)(x+3)}$?

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   $x = 1, -2$

10. Trigonometric Ratios

    State two angles, one negative and one positive, co-terminal to $\dfrac{4\pi}{5}$.

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    $\dfrac{14\pi}{5}$, $\dfrac{-6\pi}{5}$

11. Trigonometric Ratios

    State an acute angle, in quadrant 1, that is co-related to $\dfrac{7\pi}{8}$.

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    $\dfrac{3\pi}{8}$

12. Trigonometric Ratios

    Convert $680^{\circ}$ into radians as an exact value (keep as an improper fraction).

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    $\dfrac{34\pi}{9}$

13. Trigonometric Functions

    A basic sine curve has a minimum $y$ value at intervals of

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    $2 \pi$ minimum at $x = \dfrac{3\pi}{2} + 2k\pi,\ k \in \mathbb{Z}$

14. Trigonometric Functions

    The secant curve will have asymptotes at intervals of

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    $\pi$ V.A. $x = k\pi + \dfrac{\pi}{2},\ k \in \mathbb{Z}$

15. Trigonometric Functions

    The function(s) with zeros at intervals of $\dfrac{\pi}{2} + \pi n,\ n \in \mathbb{Z}$ are (full names not abbreviations or write the equations for the functions)

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    $y = \cot x$

16. Exponential & Logarithmic Functions

    Simplify the logarithm $\log_a a^n$

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    $n$

17. Exponential & Logarithmic Functions

    Simplify the logarithm $2^{\log_2(3x+4)}$

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    $3x+4$

18. Exponential & Logarithmic Functions

    The restriction on $\log_2(x+2) + \log_2(x-2)$ is

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    $x > 2$, $x \in \mathbb{R}$

19. Combining Functions

    Let $f(x) = x - 1$ and $g(x) = -2x^2 - 5$, find $x$ such that $f \circ g = g \circ f$.

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    $f \circ g = f(g(x)) = f(-2x^2-5) = (-2x^2-5)-1 = -2x^2-6$ $g \circ f = g(f(x)) = g(x-1) = -2(x-1)^2 - 5 = -2x^2 + 4x - 7$ $\therefore\ -2x^2 - 6 = -2x^2 + 4x - 7 \; , \; 4x = 1, \; x = \dfrac{1}{4}$

20. Combining Functions

    Let $f(x) = \log_2 x$ and $g(x) = x + 4$. Sketch $h(x) = (f \circ g)(x)$ and determine its domain and range.

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    $f \circ g(x) = f(x+4) = \log_2(x+4)$ $D: \{x \in \mathbb{R},\ x > -4\}$ $R: \{y \in \mathbb{R}\}$

21. Combining Functions

    Let $f(x) = mx^2 + 2x + 5$ and $g(x) = 2x^2 - nx - 2$. The functions are combined to form the new function $h(x) = f(x) \div g(x)$. Points $(1,6)$ and $(-1,10)$ satisfy the new function. Determine the values of $m$ and $n$.

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    $h(x) = \frac{mx^2 + 2x + 5}{2x^2 - nx - 2}$ $h(1) = 6 \; \dfrac{m+7}{-n} = 6 \; , \; m+7 = -6n$ $h(-1) = 10 \; \dfrac{m+3}{n} = 10 \; , \; m+3 = 10n$ $4 = -16n,\ n = \dfrac{-1}{4}$ $m = \dfrac{-11}{2}$

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