2.03 Graph quadratics
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:
State the x-intercepts.
State the y-intercept.
State 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?
State 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.
Find the x-coordinate of the vertex of the parabola represented by P \left( x \right) = p x^{2} - \dfrac{1}{2} p x - q.
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.
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.
A graph of f(x)=x^2 is shown on the right. Sketch the graph after it has undergone transformations resulting in the following functions:
g(x) = 4 x^{2} - 2
h(x) = 2 \left(x + 4\right)^{2}
Consider the two graphs. One of them has equation f(x) = x^{2} + 5.
What is the equation of the other graph?
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 parabola described by the function y = \dfrac{1}{2} \left(x - 3\right)^{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{1}{2} \left(x - 3\right)^{2}.
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.
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.
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 vertical axis of symmetry for the parabola?
Find the coordinates of the vertex of the parabola.
Is the parabola concave up or down?
Sketch the graph of y = x^{2} + x - 12.
A parabola has the equation y = x^{2} + 4 x-1.
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.
Consider the quadratic y = x^{2} - 12 x + 32.
Find the zeros of the quadratic function.
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.
Consider the function P \left( x \right) = - 2 x^{2} - 8 x + 2.
Find the coordinates of the vertex.
Sketch the graph.
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 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 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?
Using a graphing calculator, sketch the curve y = x^{2} + 6.2 x - 7.
Determine the axis of symmetry.
Determine the minimum value of y.
Use a graphing calculator to sketch 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.
Consider the function y = - 0.72 x^{2} + \sqrt{5} x + 1.21.
Find the x-coordinate of the vertex to two decimal places.
Find the y-coordinate of the vertex to two decimal places.
Is the graph shown a possible viewing window on your calculator that shows the vertex and the x-intercepts?
Use a graphing calculator to find the \\x-intercepts to two decimal places.
Consider the function y = 0.91 x^{2} - 5 x - \sqrt{5}.
Find the x-coordinate of the vertex to two decimal places.
Find the y-coordinate of the vertex to two decimal places.
Is the graph shown a possible viewing window on your calculator that shows the vertex and the x-intercepts?
Use a graphing calculator to find the \\x-intercepts to two decimal places.