 Hong Kong
Stage 4 - Stage 5

# Mixed Graphs III (quad, exp, cubic, 1/x)

Lesson

#### Worked Examples

##### question 1

Consider the quadratic function $y=-2x^2+16x-24$y=2x2+16x24.

1. Determine the $x$x-value(s) of the $x$x-intercept(s) of this parabola. Write all answers on the same line separated by commas.

2. Determine the $y$y-value of the $y$y-intercept for this parabola.

3. Determine the equation of the vertical axis of symmetry for this parabola.

4. Find the $y$y-coordinate of the vertex of the parabola.

5. Which of the following is a graph of the parabola $y=-2x^2+16x-24$y=2x2+16x24?

A

B

C

D

##### QUESTION 2

To minimise the chance of large avalanches, small explosives are fired onto mountain sides. One such explosive is fired at a height of $24$24 metres and follows a parabolic flight path. It reaches a maximum height of $122$122 metres, $180$180 metres horizontally from where it is fired.

1. Let $x$x and $y$y represent the horizontal and vertical distances of the explosive from where it is fired. Assuming the position where it is fired is $\left(0,0\right)$(0,0), determine the vertex of the parabolic path, in the form $\left(x,y\right)$(x,y).

2. Using the form of the parabola $y=-a\left(x-h\right)^2+k$y=a(xh)2+k, determine the exact value of $a$a. Then use this value to determine the equation between $x$x and $y$y.

3. The explosive hits a slope that is $361$361 metres horizontally from the point where it was fired. Does the explosive land above, at the same height or below the point where it was fired?

At the same height

A

Above

B

Below

C

##### QUESTION 3

Consider the table of values.

 number of days passed ($x$x) population of shrimp ($y$y) $1$1 $2$2 $3$3 $4$4 $5$5 $5$5 $25$25 $125$125 $625$625 $3125$3125
number of days passed ($x$x) $1$1 $2$2 $3$3 $4$4 $5$5
population of shrimp ($y$y) $5$5 $25$25 $125$125 $625$625 $3125$3125
1. Is the number of shrimp increasing by the same amount each day?

Yes

A

No

B
2. Find the equation linking population $y$y and time $x$x in the form $y=a^x$y=ax.

3. Graph the equation.

##### QUESTION 4

Consider the cubic function $y=2x^3+2$y=2x3+2

1. Is the cubic increasing or decreasing from left to right?

Increasing

A

Decreasing

B
2. Is the cubic more or less steep than the cubic $y=x^3$y=x3 ?

More steep

A

Less steep

B
3. What are the coordinates of the point of inflection of the cubic?

Inflection ($\editable{}$, $\editable{}$)

4. Plot the graph $y=2x^3+2$y=2x3+2

##### QUESTION 5

Consider the function $y=\frac{3}{x}+3$y=3x+3

1. Which of the following indicates the position of the hyperbola's branches relative to its asymptotes? A B
2. Which curve approaches positive and negative infinity more quickly, $y=\frac{1}{x}$y=1x or $y=\frac{3}{x}+3$y=3x+3?

$y=\frac{3}{x}+3$y=3x+3

A

$y=\frac{1}{x}$y=1x

B
3. What are the equations of the vertical and horizontal asymptotes?

 Vertical asymptote $x$x= $\editable{}$ Horizontal asymptote $y$y= $\editable{}$
4. Plot the graph $y=\frac{3}{x}+3$y=3x+3