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.
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.