Glossary
Axis of symmetry
A vertical line that divides a parabola into two mirror-image halves. Its equation is $$x = -b/(2a)$$.
Example:
For the function , the axis of symmetry is the line .
Downward (opening)
Describes a parabola that opens towards the negative y-direction, resembling an inverted 'U' shape. This occurs when the 'a' coefficient in standard form is negative ($$a < 0$$).
Example:
The path of a football kicked into the air forms a parabola that opens downward, indicating a maximum point.
Horizontal shifts
Transformations that move a graph left or right without altering its shape or orientation. This is achieved by adding or subtracting a constant 'h' inside the squared term ($$f(x) = (x - h)^2$$).
Example:
The graph of is a horizontal shift of the parent function 3 units to the right.
Maximum or minimum point (extrema)
The vertex of a parabola, which indicates the highest (maximum) or lowest (minimum) y-value the function can attain. This depends on the parabola's opening direction.
Example:
Finding the lowest cost to produce an item involves identifying the minimum point of the cost function.
Optimization problems
Real-world problems that involve finding the maximum or minimum value of a quantity, often modeled by quadratic functions. These problems typically ask for the 'best' possible outcome.
Example:
Determining the maximum area of a rectangular garden with a fixed perimeter is a classic optimization problem.
Parabola
The symmetrical U-shaped graph of a quadratic function. It can open either upwards or downwards.
Example:
The elegant arc of water from a garden hose forms a perfect parabola.
Parent function
The simplest form of a function type, from which other functions of the same type are derived through transformations. For quadratic functions, it is $$f(x) = x^2$$.
Example:
All parabolas are transformations of the basic parent function .
Quadratic function
A polynomial function of degree 2, characterized by an $$x^2$$ term. Its graph is always a U-shaped curve called a parabola.
Example:
The path of a ball thrown into the air can be accurately modeled by a quadratic function.
Reflections
Transformations that flip a graph across an axis. For quadratic functions, a negative sign in front of the $$x^2$$ term ($$f(x) = -x^2$$) causes a reflection across the x-axis.
Example:
The graph of is a reflection of across the x-axis, causing it to open downwards.
Standard form
The most common way to write a quadratic function, expressed as $$f(x) = ax^2 + bx + c$$, where 'a', 'b', and 'c' are constants and 'a' cannot be zero.
Example:
The equation is presented in standard form.
Upward (opening)
Describes a parabola that opens towards the positive y-direction, resembling a 'U' shape. This occurs when the 'a' coefficient in standard form is positive ($$a > 0$$).
Example:
The graph of opens upward, indicating a minimum point.
Vertex
The highest or lowest point on a parabola, located precisely on the axis of symmetry. It represents the maximum or minimum value of the function.
Example:
The peak height reached by a projectile is represented by the vertex of its parabolic trajectory.
Vertex Form
A specific form of a quadratic function, $$f(x) = a(x-h)^2 + k$$, where the coordinates of the vertex are directly given as (h,k).
Example:
The function is in vertex form, immediately showing its vertex at (-1, 4).
Vertical shifts
Transformations that move a graph up or down without altering its shape or orientation. This is achieved by adding or subtracting a constant 'k' to the function ($$f(x) = x^2 + k$$).
Example:
Adding 5 to results in a vertical shift of the parabola 5 units upwards.
Vertical stretches/compressions
Transformations that change the vertical dimension of a graph, making it narrower (stretch) or wider (compression). This occurs when the $$x^2$$ term is multiplied by a coefficient 'a' ($$f(x) = ax^2$$).
Example:
The graph of undergoes a vertical stretch, appearing narrower than .
X-intercepts (roots)
The points where the graph of a quadratic function crosses the x-axis. These are the solutions to the equation $$ax^2 + bx + c = 0$$.
Example:
If a company's profit is zero, the corresponding sales figures would be the x-intercepts of its profit function.
Y-intercept
The point where the graph of a quadratic function crosses the y-axis. It is found by setting $$x = 0$$ in the function, which simplifies to (0, c) in standard form.
Example:
In the function , the y-intercept is (0, -4).