Textbook

Advanced Functions McGraw-Hill
Chapter

Chapter 1
Section

Chapter Review Polynomial Functions

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Solutions
28 Videos

i) Which functions are polynomial functions? Justify your answer.

ii) State the degree and the leading coefficient of each polynomial function.

a) `f(x) = 3x^2 -5x + 1`

b) `g(x) = x(4-x)`

c) `h(x) = 3x + 2x`

d) `m(x) = x^{-2}`

e) `r(x) = 5(x -1)^3`

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Q1

For the graph, do the following..

- i) State whether the corresponding function has even degree or odd degree.
- ii) State whether the leading coefficient is positive or negative.
- iii) State the domain and range.
- iv) Describe the end behaviour.
- v) Identify the type of symmetry.

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Q2a

For the graph, do the following..

- i) State whether the corresponding function has even degree or odd degree.
- ii) State whether the leading coefficient is positive or negative.
- iii) State the domain and range.
- iv) Describe the end behaviour.
- v) Identify the type of symmetry.

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Q2b

Assign the following functions for below:

```
\displaystyle
\begin{array}{llll}
&y=-x^5, y = \frac{2}{3}x^4, y= 4x^3, y = 0.2x^6 \\
\end{array}
```

a) Extends from quadrant 3 to quadrant 1

b) Extends from quadrant 2 to quadrant 4

c) Extends from quadrant 2 to quadrant 1

d) Extends from quadrant 3 to quadrant 4

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Q3

Assign the following functions for below:

```
\displaystyle
\begin{array}{llll}
&g(x) = 0.5x^4 -3x^2 + 5x \\
&h(x) = x^5 -7x^3 + 2x -3\\
&p(x) = -x^6 + 5x^3 + 4
\end{array}
```

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Q4

For each polynomial function below

```
\displaystyle
\begin{array}{llll}
&g(x) = 0.5x^4 -3x^2 + 5x \\
&h(x) = x^5 -7x^3 + 2x -3\\
&p(x) = -x^6 + 5x^3 + 4
\end{array}
```

a) Determine which finite differences are constant.

b) Find the value of the constant finite differences.

c) Identify the type of symmetry, if it exists.

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Q5

a) State the degree of the polynomial function that corresponds to each constant finite difference.

- i) first differences = -5
- ii) first differences = -60
- iii) first differences = 36
- iv) first differences = 18
- v) first differences = 42
- vi) first differences = -18

b) Determine the value of the leading coefficient of each polynomial function in part a).

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Q6

Each table of values represents a polynomial function. Use finite differences to determine the following for each.

- i) the degree
- ii) the sign of the leading coefficient
- iii) the value of the leading coeffcient

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Q7a

Each table of values represents a polynomial function. Use finite differences to determine the following for each.

- i) the degree
- ii) the sign of the leading coefficient
- iii) the value of the leading coeffcient

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Q7b

A parachutist jumps from a plane 3500 m above the ground. The height, `h`

, in metres, of the parachutist above the ground t seconds after the jump can be modelled by the function `h(t) = 3500 -4.9t^2`

a) What type of function is h(t)?

b) Without calculating the finite differences, determine

- i) which finite differences are constant for this polynomial function
- ii) he value of the constant finite differences

c) Describe the end behaviour of this function assuming there are no restrictions on the domain.

d) Graph the function. State any reasonable restrictions on the domain.

e) What do the `t`

—intercepts of the graph represent for this situation?

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Q8

Use each graph of a polynomial function to determine

- i) the least possible degree and the sign of the leading coefficient
- ii) the x—intercepts and the factors of the function
- iii) the intervals where the function is positive and the intervals where it is negative

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Q9a

Use each graph of a polynomial function to determine

- i) the least possible degree and the sign of the leading coefficient
- ii) the x—intercepts and the factors of the function
- iii) the intervals where the function is positive and the intervals where it is negative

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Q9b

Sketch a graph of the polynomial function.

`y = (x+ 1)(x -3)(x + 2)`

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Q10a

Sketch a graph of the polynomial function.

`y = -x(x + 1)(x+2)^2`

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Q10b

Sketch a graph of the polynomial function.

`y = (x - 4)^2(x+3)^3`

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Q10c

The zeros of a quartic function are `-3, -1`

, and `2`

(order 2). Determine

a) equations for two functions that satisfy this condition

b) an equation for a function that satisfies this condition and passes through the point (1, 4).

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Q11

Without graphing, determine if the polynomial function has line symmetry about the y-axis, point symmetry about the origin, or neither. Graph the functions to verify your answers.

```
\displaystyle
f(x) = -x^5 . +7x^3 + 2x
```

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Q12a

Without graphing, determine if the polynomial function has line symmetry about the y-axis, point symmetry about the origin, or neither. Graph the functions to verify your answers.

```
\displaystyle
f(x) = x^4 + 3x^2 +1
```

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Q12b

Without graphing, determine if the polynomial function has line symmetry about the y-axis, point symmetry about the origin, or neither. Graph the functions to verify your answers.

```
\displaystyle
f(x) = 4x^3 -3x^2 + 8x + 1
```

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Q12c

Determine an equation for the polynomial function that corresponds to each graph.

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Q13a

Determine an equation for the polynomial function that corresponds to each graph.

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Q13b

- i) Describe the transformations that must be applied to the graph of each power function,
`f(x)`

, to obtain the transformed function. Then, write the corresponding equation. - ii) State the domain and range of the transformed function. For even functions, state the vertex and the equation of the axis of symmetry.

```
\displaystyle
f(x) =x^3, y = -\frac{1}{4}f(x) -2
```

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Q14a

- i) Describe the transformations that must be applied to the graph of each power function,
`f(x)`

, to obtain the transformed function. Then, write the corresponding equation. - ii) State the domain and range of the transformed function. For even functions, state the vertex and the equation of the axis of symmetry.

```
\displaystyle
f(x) =x^4, y = 5f[\frac{2}{5}(x -3)] + 1
```

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Q14b

- i) Write an equation for the function that results from each set of transformations.
- ii) State the domain and range. For even functions, state the vertex and the equation of the axis of symmetry.

`f(x) =x^4`

is compressed vertically by a factor of `\frac{3}{5}`

, stretched horizontally by a factor of `2`

, reflected in the y-axis, and translated 1 unit up and 4 units to the left.

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Q15a

- i) Write an equation for the function that results from each set of transformations.
- ii) State the domain and range. For even functions, state the vertex and the equation of the axis of symmetry.

`f(x) =x^4`

is compressed vertically by a factor of `\frac{3}{5}`

, stretched horizontally by a factor of 2, reflected in the y-axis, and translated 1 unit up and 4 units to the left.

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Q15b

Which are not examples of average rates of change? Explain why.

a) The average height of the players on the basketball team is 2.1 m.

b) The temperature of water in the pool decreased by 5°C over 3 days.

c) The snowboarder raced across the finish line at 60 km/h.

d) The class average on the last math test was 75%.

e) The value of the Canadian dollar increased from $0.75 U.S. to $1.01 U.S. in 8 months.

f) Approximately 30 cm of snow fell over a 5-h period.

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Q16

The graph represents the approximate value of a stock over 1 year.

a) What was the value of the stock at the start of the year? at the end of the year?

b) What does the graph tell you about the average rate of change of the value of the stock in each interval?

- i) month 0 to month 6
- ii) month 6 to month 9
- iii) month 9 to month 12

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Q17ab

The table shows the percent of Canadian households that used the Internet for electronic banking.

a) Determine the average rate of change, in percent, of households that used the Internet for electronic banking from 1999 to 2003.

b) Estimate the instantaneous rate of change in the percent of households that used the Internet for electronic banking in the year 2000, and also in 2002.

c) Compare the values you found in parts a) and b). Explain any similarities and differences.

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Q18