11.02 Parabolas (Extended)
Consider the equation y = x^{2}.
Complete the table of values:
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Use the table of values to sketch the graph of the function on a number plane.
Are the y-values ever negative?
Find the equation of the axis of symmetry.
Find the minimum y-value.
For every y-value greater than 0, how many corresponding x-values are there?
Consider the equation y = - x^{2}.
Complete the table of values:
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Use the table of values to sketch the graph of the function on a number plane.
Are the y-values ever positive?
Find the equation of the axis of symmetry.
Find the maximum y-value.
For each of the following equations:
Complete the table of values:
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Sketch the graph of the function on a number plane.
y = 3 x^{2}
y = - 2 x^{2}
y = \dfrac{1}{2} x^{2}
y = - \dfrac{1}{2} x^{2}
For each of the following equations:
Complete the table of values:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function on a number plane.
Find the minimum y-value.
Find the x-value that corresponds to the minimum y-value.
Find the coordinates of the vertex.
For each of the following equations:
Complete the table of values:
| x | - 3 | - 2 | -1 | 0 | 1 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function on a number plane.
Find the minimum y-value.
Find the x-value that corresponds to the minimum y-value.
Find the coordinates of the vertex.
For each of the following equations:
State the coordinates of the vertex.
Solve for the equation of the axis of symmetry.
Sketch the graph of the function on a number plane.
Plot the axis of symmetry.
y = \left(x - 2\right)^{2}
y = \left(x + 3\right)^{2}
Describe the transformation required to transform the parabola y = -x^{2} into the parabola y = -\left(x+4\right)^{2}.
State whether the following parabolas will be concave up or concave down:
For each of the following equations:
Complete the table of values:
| x | - 2 | - 1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function on a number plane.
State the coordinates of the vertex.
For each of the following equations:
Solve for the equation of the axis of symmetry.
Sketch the graph of the function and its axis of symmetry on a number plane.
y = -3x^{2}
y = \dfrac{1}{4}x^{2}
Describe the transformation required to transform the parabola y = x^{2} into the parabola y =5x^{2}.
Rewrite the equation, y=x^{2}, after the following transformations take place:
Graph the equation, y=x^{2}, after the following transformations take place:
Vertically translated by 4 units.
Horizontally translated by -5 units.
Vertically translated about the x axis.
Vertically scaled by 2 units.
William is training for a remote control plane aerobatics competition. He wants to fly the plane along the path of a parabola so he has chosen the equation: y = 3 x^{2} where y is the height in metres of the plane from the ground, and x is the horizontal distance in metres of the plane from its starting point.
Complete the table of values:
| x \text{ (m)} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y \text{ (m)} |
Sketch the shape of the path on a number plane.
Find the lowest height of the plane.
Find the x-value that corresponds to the minimum y-value.
Find the coordinates of the vertex.
On Jupiter the equation, d = 12.5 t^{2}, can be used to approximate the distance in metres, d, that an object falls in t seconds, if air resistance is ignored.
Complete the table of values:
| \text{time }(t) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| \text{distance } (d) |
Sketch the shape of the path on a number plane.
Use the equation to determine the number of seconds, t, that it would take an object to fall 84.4\text{ m}. Round the value of t to the nearest second.