4 Quadratics and polynomials

Worksheet

1

For the following equations:

i

Find the y-intercept.

ii

Find the x-intercepts.

iii

Complete the table of values for the equation.

iv

Find the coordinates of the turning point.

v

Sketch the graph of the parabola.

a

y = x \left(x - 4\right)

x | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

y |

b

y = \left(x - 2\right) \left(x - 6\right)

x | 0 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|

y |

c

y = \left(1 - x\right) \left(x + 5\right)

x | - 6 | - 4 | - 2 | 0 | 2 |
---|---|---|---|---|---|

y |

d

y = x \left(x + 6\right)

x | -5 | -4 | -3 | -2 | -1 |
---|---|---|---|---|---|

y |

2

For the following equations:

i

Find the y-intercept.

ii

Find the x-intercepts.

iii

Find the axis of symmetry.

iv

Find the coordinates of the turning point.

v

Sketch the graph of the parabola.

a

y = x \left(x + 4\right)

b

y = x \left(4 - x\right)

c

y = \left(x - 2\right) \left(x - 6\right)

d

y = \left(x - 3\right) \left(x + 1\right)

e

y = \left(5 - x\right) \left(x + 1\right)

f

y = \left(1 - x\right) \left(3 - x\right)

3

Consider the equation: y = \left(4 - x\right) \left(x - 20\right)

a

Find the x-intercepts.

b

Find the axis of symmetry.

c

Describe the concavity of the parabola.

d

Does the graph have a minimum or a maximum value?

e

Find the minimum or maximum y-value of the equation.

4

Consider the following graph:

a

Find the y-intercept.

b

Find the equation of the parabola in factorised form.

5

A parabola has x-intercepts at x = 1 and x = - 7.

a

Find the equation of the parabola.

b

Find the y-intercept.

c

Sketch the graph of the parabola.

6

A parabola has x-intercepts at x = \pm\sqrt{3}.

a

Find the equation of the parabola in expanded form.

b

Find the y-value of the point on the parabola where x = 1.

c

Sketch the graph of the parabola.

7

Consider the functions : f(x) = \left(x - 2\right) \left(x - 4\right) \,\,\,\,\text{and} \,\,\,\, g(x) = 2 \left(x - 2\right) \left(x - 4\right)

a

Find the y-intercept of g(x).

b

Sketch the graph of f(x) and g(x) on the same number plane.

8

Graph each pair of equations on the same number plane:

a

\text{Equation 1: } y = x \left(x + 5\right) \\ \text{Equation 2: } y = x \left(x - 3\right)

b

\text{Equation 1: } y = \left(x + 2\right) \left(x - 1\right) \\ \text{Equation 2: } y = \left(x + 4\right) \left(x - 1\right)

9

For the following equations:

i

Factorise the expression.

ii

Find the x-intercepts.

iii

Find the y-intercept.

iv

Complete the given table of values.

v

Find the coordinates of the vertex.

vi

Sketch the graph of the parabola.

a

y = 2 x + x^{2}

x | - 3 | - 2 | -1 | 0 | 1 |
---|---|---|---|---|---|

y |

b

y = x^{2} + 8 x + 12

x | - 8 | - 6 | - 4 | - 2 | 0 |
---|---|---|---|---|---|

y |

10

For the following equations:

i

Factorise the expression.

ii

Find the x-intercepts.

iii

Find the coordinates of the turning point.

iv

Sketch the graph of the parabola.

a

y = 4 x - x^{2}

b

y = x^{2} + 6 x + 8

11

For the following equations:

i

Describe the concavity of the parabola.

ii

Find the x-intercepts.

iii

Find the y-intercept.

iv

Find the axis of symmetry.

v

Find the coordinates of the vertex.

vi

Sketch the graph of the parabola.

a

y = \left(x-1\right) \left(x - 3\right)

b

y = x^{2} - 36

c

y = - x^{2} + 4

d

y = x^{2} + 4 x + 3

e

y = x^{2} + 6 x + 5

f

y = x^{2} - 8 x + 15

g

y = - x^{2} + 2 x + 24

h

y = - x^{2} - 2 x + 8

i

y = x^{2} - 2 x - 8

j

y = 8 + 2 x - x^{2}

k

y = x^{2} + 2 x - 3

l

y = 2 x^{2} + 9 x + 9

12

A football is kicked into the air and its height h metres above the ground at time t seconds after being kicked is given by :h = - t^{2} + 14 t

a

Assuming the ball starts at height 0, find the time t when it will hit the ground.

b

Find the maximum height reached by the ball.

13

The height h, in metres, reached by a ball thrown in the air after t seconds is given by the equation: h = 14 t - t^{2}

a

Complete the table of values for h = 14 t - t^{2}:

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

h |

b

Graph the relationship h = 14 t - t^{2}.

c

Find the height of the ball after 9.5 seconds have passed.

d

Find the maximum height reached by the ball.

14

When an object is thrown into the air, its height above the ground is given by the equation:

h = 163 + 38 s - s^{2}where s is its horizontal distance from where it was thrown.

a

Find s, how far horizontally the object is from where it was thrown at the point when it reaches its greatest height above the ground.

b

Find the maximum height reached by the object.

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Explore the connection between algebraic and graphical representations of relations such as simple quadratic, reciprocal, circle and exponential, using digital technology as appropriate