L.C. MATHS

## Week 10 Grinds

Mr. Kenny will go through the first of these questions during the grind. If time permits, he will then go through the second question. And so on.

Any remaining questions are left for you to practice with!

## Question 1

Graph the following cubic function:

\begin{align}f(x) = x^3-3x^2-2x+5\end{align}

in the domain $$-2\leq x \leq 4, x \in \mathbb{R}$$.

This graph will be shown during the grind!

## Question 2

(a) Graph the following cubic function:

\begin{align}f(x) = -2x^3+3x^2+5x-6\end{align}

in the domain $$-2\leq x \leq 3, x \in \mathbb{R}$$.

Using this graph, estimate the following:

(b) the roots of the equation $$f(x)=0$$

(c) the maximum turning point

(a) This graph will be shown during the grind!

(b) $$-1.5$$, $$1$$ and $$2$$

(c) $$(1.5,1.5)$$

## Question 3

(a) Graph the following exponential function:

\begin{align}f(x) = 2^x\end{align}

in the domain $$-3\leq x \leq 3, x \in \mathbb{R}$$.

Using this graph, estimate the following:

(b) $$f(1)$$

(c) $$f(2)$$

(d) $$f(1.5)$$

(a) This graph will be shown during the grind!

(b) $$2$$

(b) $$4$$

(c) $$\approx 2.8$$

## Question 4

An exponential function of the form $$y(x)=a(3^x)$$ has a $$y$$ intercept of $$(0,2)$$, where $$a\in\mathbb{R}$$.

What is the value of $$a$$?

$$2$$

## Question 5

Demonstrate that the $$y$$ intercept for all exponential functions of the form $$y(x)=a^{\pm x}$$ is $$(0,1)$$, where $$a\in\mathbb{R}$$.

Mr. Kenny will go through the first of these questions during the grind. If time permits, he will then go through the second question. And so on.

Any remaining questions are left for you to practice with!

## Theorem 13

If two triangles $$\Delta ABC$$ and $$\Delta DEF$$ are similar, then their sides are proportional, in order:

\begin{align}\frac{|AB|}{|DE|}=\frac{|BC|}{|EF|}=\frac{|AC|}{|DF|}\end{align}

The proof will be shown during the grind!

Mr. Kenny will go through the first of these questions during the grind. If time permits, he will then go through the second question. And so on.

Any remaining questions are left for you to practice with!

## Question 1

The price of five second-hand bikes is shown below.

\begin{align}\{100,80,60,40,20\}\end{align}

The first bike is one years old, the second bike is two years old etc.

(a) Represent this information on a scatter graph.

(b) What is the type of correlation?

(c) What is the strength of the correlation?

(a) This graph will be shown during the grind!

(b) Negative correlation

(c) Perfect with $$r=-1$$

## Question 2

The price of five second-hand bikes is shown below.

\begin{align}\{100,85,60,40,20\}\end{align}

The first bike is one years old, the second bike is two years old etc.

(a) Represent this information on a scatter graph.

(b) What is the type of correlation?

(c) What is the strength of the correlation?

(a) This graph will be shown during the grind!

(b) Negative correlation

(c) Strong with $$r$$ close to $$-1$$

## Question 3

The price of five second-hand bikes is shown below.

\begin{align}\{100,85,90,80,85\}\end{align}

The first bike is one years old, the second bike is two years old etc.

(a) Represent this information on a scatter graph.

(b) What is the type of correlation?

(c) What is the strength of the correlation?

(a) This graph will be shown during the grind!

(b) Negative correlation

(c) Weak with $$r$$ between $$0$$ and $$-1$$

Mr. Kenny will go through the first of these questions during the grind. If time permits, he will then go through the second question. And so on.

Any remaining questions are left for you to practice with!

## Question 1

Consider the following sequence:

\begin{align}3,6,12,24…\end{align}

(a) Show that this sequence is geometric.

(b) What is the next term in this sequence?

(c) What is the general term for this sequence?

(d) What is the $$10$$th term in this sequence?

(a) Successive terms have a common ratio of $$2$$.

(b) $$48$$

(c) $$T_n=3(2^{n-1})$$

(d) $$536$$

## Question 2

Consider the following sequence:

\begin{align}4,8,16,32,…\end{align}

(a) Show that this sequence is geometric.

(b) What is the general term for this sequence?

(c) What is the sum of the first $$10$$ terms of this sequence?

(a) Successive terms have a common ratio of $$2$$.

(b) $$T_n=4(2^{n-1})$$

(c) $$4{,}092$$

## Question 3

€$$100$$ is invested into a bank every month for $$2$$ years at a monthly rate of $$0.5\%$$.

What is the total amount at the end of the two years?

€$$2{,}555.91$$

## Question 4

€$$300$$ is invested into a bank every three months month for $$6$$ years at a quarterly rate of $$1\%$$.

What is the total amount at the end of the six years?

(a) Successive terms have a common ratio of $$2$$.

(b) $$T_n=4(2^{n-1})$$

(c) $$4{,}092$$