Vertex & Axis of Symmetry
What is the highest or lowest point on a parabola called?
The Vertex
The points where a parabola intersects the x-axis.
x-intercepts
What is the domain of all quadratic functions?
All real numbers
How does changing the value of 'a' in the equation y = a(x - h)² + k affect the shape of the parabola?
It affects the width and direction of the opening.
How can quadratic functions be used to model the path of a projectile? (Define both variables)
The height of the projectile over time can be modeled by a quadratic function.
The vertical line that divides a parabola into two symmetrical halves.
The Axis of Symmetry
The point where a parabola intersects the y-axis.
y-intercepts
If a parabola opens upward, is the range limited to values greater than or less than the y-coordinate of the vertex?
Greater than or equal to
What transformation occurs when the equation of a parabola is changed from y = x² to y = x² + 5?
Vertical shift 5 units upward
A company wants to maximize its profit. How can quadratic functions help them determine the optimal price for their product?
Profit can often be modeled as a quadratic function of the price.
Find the vertex of the parabola represented by the equation: y = (x - 3)² + 2
(3, 2)
If a parabola has no x-intercepts, what does that tell you about the solutions to the related quadratic equation?
The solutions are imaginary.
A parabola opens downward and has a vertex at (1, -3). What is the range of the function?
y ≤ -3
Describe the transformation that occurs when the equation of a parabola is changed from y = x² to y = (x - 3)².
Horizontal shift 3 units to the right
Describe how quadratic functions can be used in architecture and design. (HINT: think about real world examples shaped like parabolas)
They can be used to design arches, bridges, and other structures.
If the axis of symmetry of a parabola is x = -5, what is the x-coordinate of the vertex?
-5
What are the other three names that mean the same thing as x-intercepts?
Roots, solutions, zeroes
Explain the difference between domain and range.
Domain refers to the possible input values (x-values), while range refers to the possible output values (y-values).
Given y = -2(x+3)2 - 5, identify the transformations that have been applied to the parent function y = x².
Reflection, stretch by 2, left 3, down 5
A ball is thrown into the air. The height of the ball, in feet, can be modeled by the equation h(t) = -16t² + 48t, where t is the time in seconds. Find the maximum height of the ball.
The maximum height is 36 feet.
Given the graph of a parabola, how can you visually determine the axis of symmetry?
Find the vertical line that passes through the vertex.
A parabola has x-intercepts at (-2, 0) and (4, 0). What is the equation of the axis of symmetry?
x=1
An object is launched and the height (ft) after time (s) has passed can be represented by the equation
y = -4.9x2+19.6x+5.75
Determine the domain and range in the context of the problem.
Domain: [0, 4.275]
Range: [0, 25.35]
A parabola has been reflected across the x-axis, vertically compressed by 1/3, and shifted 2 units to the left. Write a possible equation for the parabola.
y = -1/3(x+2)2
A farmer wants to enclose a rectangular field using a river as one side. He has 1000 meters of fencing. Find the dimensions of the field that will maximize the area.
width = 250 m
length = 500 m