## H1011 Quizzes

Quiz 1: Elementary functions
Question 1 Questions
Find the gradient and the $y$-intercept of the straight line through the points (4,3) and (7,5).
 a) gradient $\frac{3}{2}$ and y-intercept $\frac{1}{3}$. b) gradient $\frac{2}{3}$ and y-intercept $\frac{1}{3}$. c) gradient $\frac{3}{2}$ and y-intercept $-\frac{1}{2}$. d) gradient $\frac{2}{3}$ and y-intercept $-\frac{1}{2}$.

Choice (a) is incorrect
Choice (b) is correct!
$m=\frac{5-3}{7-4}=\frac{2}{3}$.
$\text{therefore,}y=\frac{2}{3}x+b⇒3=\frac{8}{3}+b⇒b=\frac{1}{3}$.
Choice (c) is incorrect
Choice (d) is incorrect
Find the equation of the straight line through (1,1) and (8,-2).
 a) $3x-7y=10$ b) $3x+7y=-10$ c) $3x+7y=10$ d) $3x-7y=-10$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Gradient = $-\frac{3}{7}$, y-intercept=$\frac{10}{7}$.
Choice (d) is incorrect
The equation of the line through (1,1) and (3,4) is
 a) $3y-2x-1=0$ b) $2y-3x-1=0$ c) $3x-2y-1=0$ d) $2y+3x-1=0$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$m=\frac{4-1}{3-1}=\frac{3}{2}$
$\text{therefore,}y=\frac{3}{2}x+b⇒1=\frac{3}{2}+b⇒b=\frac{1}{2}$
$\text{therefore,}y=\frac{3}{2}x-\frac{1}{2}⇒2y=3x-1⇒3x-2y-1=0.$
Choice (d) is incorrect
Which of the following is the graph of $y=-2{\left(x+1\right)}^{2}+4$?
 a) b) c) d)

Choice (a) is incorrect
Not correct. This is the graph of $2{\left(x+1\right)}^{2}+4$.
Choice (b) is incorrect
Not correct. This is the graph of $-2{\left(x-1\right)}^{2}+4$.
Choice (c) is correct!
Choice (d) is incorrect
Not correct. This is the graph of ${\left(x+1\right)}^{2}+4$.
Which of the following expressions is equivalent to $-3{x}^{2}-3x+7$?
 a) $-{\left(3x+\frac{3}{2}\right)}^{2}+\frac{5}{2}$ b) $-3{\left(x-\frac{1}{2}\right)}^{2}+\frac{31}{4}$ c) $-3{\left(x+\frac{1}{2}\right)}^{2}+\frac{31}{4}$ d) $-3{\left(x+1\right)}^{2}+10$ e) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
 $\begin{array}{cc}\begin{array}{rl}-3{x}^{2}-3x+7& =-3\left({x}^{2}+x\right)+7\\ & =-3\left({x}^{2}+x+{\left(\frac{1}{2}\right)}^{2}-{\left(\frac{1}{2}\right)}^{2}\right)+7\\ & =-3{\left(x+\frac{1}{2}\right)}^{2}+\frac{3}{4}+7\\ & =-3{\left(x+\frac{1}{2}\right)}^{2}+\frac{31}{4}\end{array}& \end{array}$
Choice (d) is incorrect
Choice (e) is incorrect
Which of the following functions describes the above graph?
 a) $y=-{\left(x-1\right)}^{2}+2$ b) $y=2x\left(2-x\right)$ c) $y=-2{\left(x+1\right)}^{2}+2$ d) $y=2{\left(x-1\right)}^{2}+2$ e) None of the above.

Choice (a) is incorrect
Incorrect answer. The given graph is “taller“ or “skinnier“ than the graph of $y=-{\left(x-1\right)}^{2}+2$. You can see that this answer is incorrect by plotting some points.
Choice (b) is correct!
Correct. This expression is equal to $y=-2{\left(x-1\right)}^{2}+2$.
Choice (c) is incorrect
Incorrect answer. This function has its turning point at x=-1
Choice (d) is incorrect
Incorrect answer. The graph of this function will point upwards.
Choice (e) is incorrect
Simplify ${e}^{2x}{e}^{5{x}^{2}}$.
 a) ${e}^{7{x}^{2}}$ b) ${e}^{10{x}^{3}}$ c) ${e}^{x\left(2+5x\right)}$ d) ${e}^{2x}+{e}^{5{x}^{2}}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
${e}^{2x}{e}^{5{x}^{2}}={e}^{2x+5{x}^{2}}={e}^{x\left(2+5x\right)}.$
Choice (d) is incorrect
Simplify $ln\left(\frac{a{b}^{3}}{{c}^{2}}\right)$.
 a) $lna+ln3b-ln2c$ b) $lna+3lnb-2lnc$ c) $3lnab-2lnc$ d) $3lnab-ln2c$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{rcll}ln\left(\frac{a{b}^{3}}{{c}^{2}}\right)& =& lna{b}^{3}-ln{c}^{2}& \text{}\\ & =& lna+ln{b}^{3}-2lnc& \text{}\\ & =& lna+3lnb-2lnc.& \text{}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Which of the following is the graph of $y=-{e}^{\left(x-2\right)}+1$?
 a) b) c) d) e) None of the above

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which of the following graphs represents the function $y={\left(x+2\right)}^{-1}+\pi$?
Incorrect answer. You need to move this graph up $\pi$ units. $\pi$ is approximately equal to 3.14
Incorrect answer. Hint: ${\left(x+2\right)}^{-1}=\frac{1}{x+2}$.
Incorrect answer. The given function has a vertical aymptote at $x+2=0$, i.e. $x=-2$. Hint: ${\left(x+2\right)}^{-1}=\frac{1}{x+2}$