Difference between revisions of "Week2a"

• Handout on linear functions
• Handout on quadratic functions
• Handout on various functions

Activity: working with linear functions.

The temperature scales Celsius and Fahrenheit have a linear relationship with $0^{\circ}C = 32^{\circ}F$ and $100^{\circ}C = 212^{\circ}F$.

1. Write a linear equation expressing $F$ in terms of $C$ and sketch its graph.
2. If it is $75^{\circ}C$, what is the temperature in $F$?
3. If it is $75^{\circ}F$, what is the temperature in $C$?
4. What about if it is $175^{\circ}F$?
5. What about if it is $-25^{\circ}F$?
6. Did you compute each $C$ in parts (3) - (5) separately? Could you have avoided this?
7. Graph the line $F=C$ on the same axes. Can you tell from your graph if there a temperature $x$ for which $x^{\circ}C=x^{\circ}F$? If so, what is $x$?

Activity: working with quadratic functions.

The revenue of Miley Cyrus' concert in Vancouver depends on the number of unsold seats. Miley's manager calculates that the revenue is given by $R(x) = 8000+70x-x^2$, where $x$ is the number of unsold seats. Find the maximum revenue and the number of unsold seats that corresponds to maximum revenue.

Activity: working with general functions.

Consider the following table of function values for $f(x)$ and $g(x)$.

a) Evaluate: $(f+g)(4)$, $\dfrac{f}{g}(3)$, $\dfrac{g}{f}(3)$, $(f\cdot g)(5)$, $(f\circ g)(5)$, $(g\circ f)(2)$, $f^{-1}(-3)$, $g^{-1}(8)$.

b) Suppose $f$ is even and $g$ is odd. Evaluate $f(-3)$ and $g(-5)$.

Extra discussion/group work questions

1. Find a constant $k$ such that the line through the points $(k,2)$ and $(3,6)$ has the same rate of change as the line $y=4x+3$.
2. Graphically determine the $x$-values for which $\frac{2}{5}x+7>3$.
3. Suppose that a bar charges a $\$8$ cover fee upon entry and $\$4.75$ per drink. Write a linear equation representing your cost for a night out in terms of how many drinks you ordered.
4. Find the number $c$ such that $x^2+5x+c$ is a perfect square for all $x$ values.
5. Determine the $x$-values for which $x^2+4x+1\geq 1$. How would you solve this graphically?

Question.

What is the domain of $f(x)=\sqrt{x-5}$?

a) $x\geq 5$.

b) $x> 5$.

c) $x\neq 5$.

d) All real numbers.

Question.

What is the domain of $f(x)=\dfrac{2x}{\sqrt{x-5}}$?

a) $x\geq 5$.

b) $x> 5$.

c) $x\neq 5$.

d) All real numbers.

Question.

Which of the following functions has its domain equal to its range?

a) $f(x)=x^2$.

b) $g(x)=\sqrt{x}$.

c) $h(x)=|x|$.

d) All functions have that property.

Question.

What is the slope of a line passing through the points $(1,4)$ and $(3,8)$?

a) $1/2$.

b) $-1/2$.

c) $2$.

d) $-2$.

Question.

Which of the following lines has a different slope than all the others?

a) $6x-2y+7=0$.

b) $y=3x-1$.

c) $3y=3x-4$.

d) $9x=3y+3$.

Question.

What is the average rate of change of $f(t)=t^2$ between $t=1$ an $t=3$?

a) 1/8.

b) 1/4.

c) 4.

d) 8.

Question.

Consider the following table of function values for $f(x)$ and $g(x)$.

What is $f(g(5))$?

a) 4.

b) 7.

c) 34.

d) Cannot be determined.

Question.

Consider the following table of function values for $f(x)$ and $g(x)$.

If $g(f(x))=9$, then what is $x$?

a) 0.

b) 1.

c) 2.

d) Cannot be determined.

Question.

Consider the following table of function values for $f(x)$ and $g(x)$.

What is $f(g(5))$?

a) 4.

b) 7.

c) 34.

d) Cannot be determined.