Further problems 3

Exercise 1

Q1

Draw all the points $P(x|y)$ with $y\leq 2x+1$ in a 2d-coordinate system.

Hint: first, indicate the points with y=2x+1, and then try out other points on both side of this curve.

Q2

The four graphs below describe a running competition between four people.

What is the distance they had to run?
Create a ranking.
Assign each graph to one of the following statements:
- takes a break in the middle of the race.
- starts quick and slows down towards the end.
- starts slow and increases the speed.
- runs with constant speed.

Q3

The vessels are filled with water from a tap. The height of the water is measured as a function of time and a graph is produced. Which graph corresponds to which vessel?

Q4

Consider the function $f(x)=\sqrt{x}$ .

The point $A$ is on the graph of $f$ and has the $x$ -coordinate $8$ . Determine its $y$ -coordinate.
The point $B$ is on the graph of $f$ and has the $y$ -coordinate $27$ . Determine its $x$ -coordinate.

Q5

Consider the functions $f(x)=2x^2+3x$ and $g(x)=-x^2$

Draw the graphs into the same coordinate system.
Calculate $f(1)$ and $f(-3)$ .
Calculate the $x$ - and $y$ -intercepts of $f$ and $g$ .
Determine the points of intersections between $f$ and $g$ .

Q6

Shown below is the graph of a function $f$ .

Based on the graph, estimate $f(-3)$ and $f(0)$ .
Based on the graph, find an $x$ with $f(x)=1$ .
Based on the graph, the $x$ - and $y$ -intercept.
Determine $f(200)$ . Hint: Find the function equation first.

Solution

A1

Shaded area, including the straight line.

A2

$400m$
D (fastest), B, A, C
It is
- C: takes a break in the middle of the race.
- A: starts quick and slows down towards the end.
- D: starts slow and increases the speed.
- B: runs with constant speed.

A3

a: IV, b: I , c: III, d: II

A4

$y=f(8)=\sqrt{8}=\underline{2.828...}$
Find $x$ with $f(x)=27$ , that is, with $\sqrt{x}=27$ . Squaring both sides, we get $x=27^2=\underline{729}$

A5

The graphs are shown below.
$f(1)=2\cdot 1^2+3\cdot 1=\underline{5}$ , $f(-3)=2\cdot (-3)^2+3\cdot (-3)=\underline{9}$
$x$ -intercepts of $f$ : find $x$ with $f(x)=2x^2+3x=0$ We get $x_1=\underline{0}$ and $x_2=\underline{-1.5}$ . $y$ -intercept of $f$ : $y=f(0)=\underline{0}$ . $x$ -intercepts of $g$ : find $x$ with $g(x)=-x^2=0$ It follows $x=\underline{0}$ . $y$ -intercept of $g$ : $y=g(0)=\underline{0}.$
To find the $x$ -coordinate of the intersection points $A$ and $B$ , find $x$ with $f(x)=g(x)$ . So solve the equation $\begin{array}{lll} 2x^2+3x &=& -x^2 & \quad \vert +x^2\\ 3x^2+3x &=& 0 & \quad \vert \text{factor out}\\ 3x(x+1) &=& 0 \end{array}$ It follows $x_1=0$ , and $x_2=-1$ . The corresponding $y$ -coordinates are $y_1=f(x_1)=0$ and $y_2=f(x_2)=-1$ . Thus, the intersection points are $\underline{A(0|0)}$ and $\underline{B(-1|-1)}$ .

A6

$f(-3)=\underline{-2}, f(0)=\underline{-1}$
$x=\underline{6}$
$x$ -intercept is intersection with $x$ -axis, so $x=\underline{3}$ . $y$ -intercept is intersection with $y$ -axis, so $y=\underline{-1}$ .
Create a table of values to find the rule:
$\begin{array}{r|r} x & y \\\hline -3 & -2\\ 0 & -1\\ 3 & 0\\ 6 & 1\\ \end{array}$
Try some equations ... one that works is $f(x)=\frac{1}{3}x -1$ . So $f(200)=\frac{1}{3}\cdot 200-1=\underline{65.\overline{6}}$ .