21.05 Quadratic functions
Consider the general quadratic equation y = a x^{2} + b x + c, a \neq 0.
If a \lt 0, in what direction will the parabola open?
If a \gt 0, in what direction will the parabola open?
Does the parabola represented by the equation y = x^{2} - 8 x + 9 open upward or downward?
Does the graph of y = x^{2} + 6 have any x-intercepts? Explain your answer.
State whether the following parabolas have x-intercepts:
Consider the given graph:
What are the x-intercepts?
What is the y-intercept?
What is the maximum value?
Consider the given graph:
Is the curve concave up or concave down?
State the y-intercept of the graph.
What is the minimum value?
At which value of x does the minimum value occur?
Determine the interval of x for which the graph is decreasing.
Consider the graph of the parabola:
State the coordinates of the x-intercept.
State the coordinates of the vertex.
State whether the following statements are true about the vertex:
The vertex is the minimum value of the graph.
The vertex occurs at the x-intercept.
The vertex lies on the axis of symmetry.
The vertex is the maximum value of the graph.
Suppose that a particular parabola is concave down, and its vertex is located in quadrant 2.
How many x-intercepts will the parabola have?
How many y-intercepts will the parabola have?
Suppose that a particular parabola has two x-intercepts, and its vertex is located in quadrant 4. Will such a parabola be concave up or concave down?
Consider the quadratic function defined in the table on the right:
What are the coordinates of the vertex?
What is the minimum value of the function?
| x | y |
|---|---|
| -7 | 11 |
| -6 | 6 |
| -5 | 3 |
| -4 | 2 |
| -3 | 3 |
| -2 | 6 |
| -1 | 11 |
A vertical parabola has an x-intercept at \left(-1, 0\right) and a vertex at \left(1, - 6 \right). Find the other \\x-intercept.
State whether the following can be found, without any calculation, from the equation of the form y = \left(x - h\right)^{2} + k but not from the equation of the form y = x^{2} + b x + c:
x-intercepts
y-intercept
vertex
Quadratic function A is represented graphically as shown. Quadratic function B, which is concave down, shares the same x-intercepts as quadratic function A, but has a y-intercept closer to the origin. Which of the functions has a greater maximum value?
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 equation y = 25 - \left(x + 2\right)^{2}. What is the maximum value of y?
Consider the function y = \left(14 - x\right) \left(x - 6\right).
State the zeros of the function.
Find the axis of symmetry.
Is the graph of the function concave up or concave down?
Determine the maximum y-value of the function.
Consider the parabola of the form y = a x^{2} + b x + c, where a \neq 0.
Complete the following statement:
The x-coordinate of the vertex of the parabola occurs at x = ⬚. The y-coordinate of the vertex is found by substituting x = ⬚ into the parabola's equation and evaluating the function at this value of x.
Consider the graph of the function
f \left( x \right) = - x^{2} - x + 6:
Using the graph, write down the solutions of the equation - x^{2} - x + 6 = 0.
True or false:
The quadratic formula can be used to find the y-intercept.
If the parabola has only one x-intercept , then the x-intercept is also the vertex.
Consider the parabola whose equation is y = 3 x^{2} + 3 x - 7. Find the x-intercepts of the parabola in exact form.
Consider the parabola described by the function y = - 2 x^{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 = - 2 x^{2} + 2.
Consider the two graphs. One of them has equation f(x) = -x^{2} + 5.
What is the equation of the other graph?
Use the graph to approximate the solutions of the equation -x^2+5=0 to one decimal place.
Consider the quadratic function h \left( x \right) = x^{2} + 2.
Sketch the graph of the parabola h \left( x \right).
Plot the axis of symmetry of the parabola on the same graph.
What is the vertex of the parabola?
Consider the equation y = \left(x - 3\right)^{2} - 1.
Find the x-intercepts.
Find the y-intercept.
Determine the coordinates of the vertex.
Sketch the graph.
Use your graph to find the solutions of the equation \left(x - 3\right)^{2} - 1=0.
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 of this parabola?
What is the y-coordinate of the graph of f \left( x \right) at x = -1?
Sketch the graph of the parabola.
Plot the axis of symmetry of the parabola on the same graph.
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 parabola y = \left(2 - x\right) \left(x + 4\right).
State the y-intercept.
State the x-intercepts.
Complete the table of values:
Determine the coordinates of the vertex of the parabola.
Sketch the graph of the parabola.
| x | -5 | -3 | -1 | 1 | 3 |
|---|---|---|---|---|---|
| y |
Consider the parabola y = \left(x - 3\right) \left(x - 1\right).
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.
Consider the parabola y = x \left(x + 6\right).
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.
Sketch the graph of the following:
Consider the function y = \left(x + 5\right) \left(x + 1\right).
Sketch the graph.
Sketch the graph of y = - \left(x + 5\right) \left(x + 1\right) on the same set of axes.
Consider the equation y = x^{2} - 6 x + 8.
Factorise the expression x^{2} - 6 x + 8.
Hence, or otherwise, find the x-intercepts of the quadratic function y = x^{2} - 6 x + 8
Find the coordinates of the turning point.
Sketch the graph of the function.
Consider the parabola y = x^{2} + x - 12.
Find the x-intercepts of the curve.
Find the y-intercept of the curve.
What is the equation of the vertical axis of symmetry for the parabola?
Find the coordinates of the vertex of the parabola.
Sketch the graph of y = x^{2} + x - 12.
A parabola has the equation y = x^{2} + 4 x-1.
Express the equation of the parabola in the form y = \left(x - h\right)^{2} + k by completing the square.
Find the y-intercept of the parabola.
Find the vertex of the parabola.
Is the parabola concave up or down?
Hence, sketch the graph of y = x^{2} + 4 x-1.
Use your graph to approximate the solutions of the equation x^{2} + 4 x-1=0.
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 by completing the square.
Find the coordinates of the vertex of the parabola.
Hence, sketch the graph.
Consider the curve y = x^{2} + 6 x + 4.
Determine the axis of symmetry.
Hence, determine the minimum value of y.
Sketch the graph of the function.
Use your graph to approximate the solutions of the equation x^{2} + 6 x + 4=0.
Consider the function P \left( x \right) = - 2 x^{2} - 8 x + 2.
Find the coordinates of the vertex.
Sketch the graph.
Use your graph to approximate the solutions of the equation - 2 x^{2} - 8 x + 2=0.
Consider the equation y = 6 x - x^{2}.
Find the x-intercepts of the quadratic function.
Find the coordinates of the turning point.
Sketch the graph.
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 for this curve.
Find the axis of symmetry.
Find the y-coordinate of the vertex of the parabola.
Sketch the graph.
Use your calculator or other handheld technology to graph the equations below. Then answer the following questions:
What is the vertex of the graph?
y = 4 x^{2} - 64 x + 263
y = - 4 x^{2} - 48 x - 140
Use your calculator or other handheld technology to graph y = - 3 x^{2} - 12.
What is the vertex of the graph?
Are there any x-intercepts?
For what values of x is the parabola decreasing?
Use technology to graph the parabola y = - 2 x^{2} + 16 x - 24.
Find the x-intercepts of the parabola.
Find the y-intercept of the parabola.
Find the axis of symmetry of the parabola.
Find the y-coordinate of the vertex of the parabola.