Is the function y = - 2 \left(x - 3\right)^{2} - 1 a one-to-one function?
Consider the quadratic equation y = - x^{2} + 10 x + 16.
Find the y-intercept of the graph given by the equation.
Consider the graph of the given function y = f \left( x \right):
What is the minimum value of the graph?
What is the range of the function?
What is the domain of this function?
Consider the graph of the quadratic equation.
State the y-intercept.
State the x-intercepts.
State the equation of the axis of symmetry.
State the coordinates of the maximum point.
Determine the values of x for which the gradient of the curve is negative.
Find the maximum value of y for the quadratic function y = - x^{2} + 10 x - 25.
The minimum value of the function y = 9 x^{2} + 108 x + m is y = 6.
Find the value of x at the minimum point.
Find the value of m.
Determine the values of x for which the quadratic function is decreasing.
Consider the parabola defined by the equation y = x^{2} + 5.
Is the parabola concave up or concave down?
What is the y-intercept of the parabola?
What is the minimum y-value of the parabola?
Hence determine the range of the parabola.
For each of the following equations of parabolas:
Determine if the parabola is concave up or concave down.
Find the y-intercept of the parabola.
Find the number of solutions of y=0.
Find the number of x-intercepts.
Find the minimum y-value of the parabola.
y = x^{2} + 3
y = \left(x - 3\right)^{2} + 2
Consider the equation y = 25 - \left(x + 2\right)^{2}.
What is the maximum value of y?
State whether the following parabolas have any x-intercepts:
y = \left(x - 7\right)^{2} + 4
y = - \left(x - 7\right)^{2} + 4
y = - \left(x - 7\right)^{2} - 4
y = \left(x - 7\right)^{2} - 4
What is the axis of symmetry of the parabola y = k \left(x - 7\right) \left(x + 7\right) for any value of k?
Consider the following graph:
Find the x-intercepts.
Find the zeros of the function.
Consider the graph of the function f \left( x \right) = - x^{2} - x + 6.
Using the graph, write down the solutions to the equation - x^{2} - x + 6 = 0.
Use the quadratic formula to find the x-intercepts of y = 3 x^{2} + 3 x - 7 in exact form.
The parabola y = 2 x^{2} + b x + 1 has its axis of symmetry at x = - 1. Find the value of b.
For the graph y = x^{2}, find the two x-values that correspond to a y-value of 81.
Determine the value of c if the parabola y = x^{2} + 4 x + c has exactly one x-intercept.
Consider the given graph of the parabola:
State the x-intercepts.
State the y-intercept.
Find the equation of the axis of symmetry.
State the coordinates of the vertex.
State whether the following statements are true about the vertex:
The x-value of the vertex is the average of the x-values of the two x-intercepts.
The vertex is the minimum value of the graph.
The vertex is the maximum value of the graph.
The vertex lies on the axis of symmetry.
The y-value of the vertex is the same as the y-value of the y-intercept.
Consider the table of values generated from a quadratic function:
| x | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|
| y | 23 | 13 | 7 | 5 | 7 | 13 | 23 |
What are the coordinates of the vertex?
Is the vertex a maximum or minimum point?
What is the minimum value of the function?
Find the coordinates of the vertex of y = 3 x^{2} - 6 x - 9.
A parabola has an x-intercept at \left(-1, 0\right) and a vertex at \left(1, - 6 \right). Find the coordinates of the other x-intercept.
Consider the parabola defined by equation y = 2 \left(x - \dfrac{3}{4}\right)^{2} - \dfrac{1}{4}.
Write down the coordinates of the vertex.
Find the vertical axis of symmetry for this parabola.
Write down the coordinates of the y-intercept.
Consider the function y = x^{2} - 4 x + 6.
Find the coordinates of the vertex.
Is the vertex the maximum or minimum of this parabola?
How many real roots does this parabola have?
Consider the quadratic function h \left( x \right) = x^{2} - 2.
Sketch the graph of the parabola given by h \left( x \right).
Sketch the axis of symmetry of the parabola on the same axes.
What is the vertex of this parabola?
Consider the parabola described by the function y = \dfrac{x^{2}}{2} - 2
Is the parabola concave up or down?
Is the parabola more or less steep than the parabola y = x^{2}?
What are the coordinates of the vertex of the parabola?
Sketch the graph of y = \dfrac{x^{2}}{2} - 2.
A parabola of the form y = a x^{2} goes through the point \left(2, - 4 \right).
What is the value of a?
What are the coordinates of the vertex?
Sketch the graph of the parabola.
Consider the quadratic function f \left( x \right) = \left(x - 3\right)^{2}.
What are the coordinates of the vertex of this parabola?
What is the equation of the axis of symmetry of this parabola?
Graph the parabola corresponding to f \left( x \right).
Sketch the axis of symmetry of the parabola on the same axes.
For each of the following parabolas:
Find the y-intercept.
Find the x-intercepts.
State the equation of the axis of symmetry.
Find the coordinates of the turning point.
Sketch the graph of the parabola.
y = x \left(x + 6\right)
y = x \left(6 - x\right)
y = \left(2 - x\right) \left(4 - x\right)
Graph the following quadratic functions:
Consider the quadratic function y = 16 - x^{2}.
Find the x-intercepts.
Find the y-intercept.
Sketch the graph of the function.
State the vertex of the parabola.
The equation of a parabola is of the form y = \left(x - a\right) \left(x - b\right).
The parabola has x-intercepts x = \sqrt{3} and x = - \sqrt{3}. Write down its equation in expanded form.
What is the y-value of the point on the parabola where x = 1?
Sketch the graph of the curve.
Consider the parabola y = x^{2} + x - 12.
Factorise the equation of the parabola.
Find the x-intercepts of the curve.
Find the y-intercept of the curve.
What is the equation of the axis of symmetry?
Find the coordinates of the vertex.
Is the parabola concave up or down?
Sketch the curve of y = x^{2} + x - 12.
Consider the quadratic function y = - x^{2} + 4 x + 12.
What is the concavity of the parabola?
What is the y-intercept?
Find the x-intercepts.
Find the equation of the axis of symmetry.
Find the coordinates of the vertex.
Sketch the graph of the function y = - x^{2} + 4 x + 12.
A parabola is described by the function y = 2 x^{2} + 9 x + 9.
Find the x-intercepts of the parabola.
Find the y-intercept of this curve.
Find the equation of the axis of symmetry.
Find the y-coordinate of the vertex.
Sketch the curve of y = 2 x^{2} + 9 x + 9.
Consider the quadratic function f \left( x \right) = - 3 \left(x + 2\right)^{2} - 4.
What are the coordinates of the vertex of this parabola?
What is the equation of the axis of symmetry?
What is the y-coordinate of the graph of f \left( x \right) at x = -1?
Sketch the graph of the parabola f \left( x \right).
Sketch the axis of symmetry of the parabola on the same axes.
Consider the equation y = \left(x - 3\right)^{2} - 1.
Find the x-intercepts.
Find the y-intercept.
Find the coordinates of the vertex.
Sketch the graph of the equation.
Consider the parabola described by the function y = - \dfrac{1}{5} \left(x - 1\right)^{2} + 1.
Is the parabola concave up or down?
Is the parabola more or less steep than the parabola y = - x^{2}?
What are the coordinates of the vertex?
Sketch the graph of the parabola.
Consider the quadratic y = x^{2} - 12 x + 32
Find the zeros of the quadratic function.
Express the equation in the form y = a \left(x - h\right)^{2} + k.
Find the coordinates of the vertex.
Sketch the graph of the parabola.
A parabola of the form y = \left(x - h\right)^{2} + k is symmetrical about the line x = 2, and its vertex lies 6 units below the x-axis.
Write the equation of the parabola.
Sketch the graph of the parabola.
On a number plane, sketch the shape of a parabola of the form y = a \left(x - h\right)^{2} + k that has the following signs for a, h and k:
Consider the two curves shown. The top curve has equation f(x) = x^{2} + 5.
State the equation of the g(x).
Consider a parabola whose x-intercepts are - 10 and 4, and whose y-intercept is - 40. Find the equation of the parabola.
Consider the following graph:
What is the y-intercept of the graph?
Write the equation of this graph in factored form.
A parabola has its turning point at x = - 3 and one of the x-intercepts at x = 1.
What is the other x-intercept?
If it has a y-intercept at 3, state the equation of the parabola.
What are the coordinates of the turning point?
Consider the following graph:
Determine the coordinates of the vertex.
Find the equation of the parabola.
A parabola is of the form y = \left(x - h\right)^{2} + k. It has x-intercepts at \left(1, 0\right) and \left( - 5 , 0\right).
Determine the axis of symmetry of the curve.
Hence or otherwise find the equation of the curve.
Suppose a parabola has its vertex at \left( - 1 , - 8 \right) and coefficient a = \dfrac{6}{5}.
Write the equation of the parabola in vertex form.
Write the equation of the parabola in general form.
A family of quadratics is defined as having a vertex of \left(1, 3\right).
Letting a be the coefficient of x^{2}, write the equation that represents this family of quadratics.
Find the equation of the quadratic in this family that has a vertex at \left(1, 3\right) and passes through the point \left(9, 5\right).
Find the equation of the quadratic in this family that has x = 5 as one of its zeros.
Find the equation of the quadratic function that has a vertex at \left( - 12 , 3\right) and that passes through the point \left( - 4 , 19\right).
Over the summer, Susana and her friends build a bike ramp to launch themselves into a lake. Susana decides that the shape of the ramp will be parabolic, and that the best parabola is given by the equation y = \dfrac{1}{4} \left(x + 2\right)^{2} + 2, where y is the height in metres above the ground, and x is the horizontal distance in metres from the edge of the lake.
If the ramp starts 6\text{ m} back from the edge of the lake, where x = -6, how high is the start of the ramp?
At what height is the end of the ramp?
At what other distance x is the rider also at this height?
