Key Concepts The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down. The axis of symmetry is the vertical line passing through the vertex. Quadratic functions are often written in general form.

Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Interpreting a function means converting the symbols of a formula or a drawn graph into meaningful information that fits what you’re looking for. When you’re interpreting a function, you’re answering questions based on the occasionally cryptic information available.

Exponential Function Properties

• The domain is all real numbers.
• The range is y>0.
• The graph is increasing.
• The graph is asymptotic to the x-axis as x approaches negative infinity.
• The graph increases without bound as x approaches positive infinity.
• The graph is continuous.
• The graph is smooth.

The domain of exponential functions is all real numbers. The range is all real numbers greater than zero. The line y = 0 is a horizontal asymptote for all exponential functions. When a > 1: as x increases, the exponential function increases, and as x decreases, the function decreases.

How can we tell if a function is increasing or decreasing?

1. If f′(x)>0 on an open interval, then f is increasing on the interval.
2. If f′(x)<0 on an open interval, then f is decreasing on the interval.

Every exponential function is strictly increasing.

This graph is decreasing, but all the function values are negative. The range for an exponential function is always positive values. This graph is increasing, but all the function values are negative.

It gets rapidly smaller as x increases, as illustrated by its graph. In the exponential growth of f(x), the function doubles every time you add one to its input x. The presence of this doubling time or half-life is characteristic of exponential functions, indicating how fast they grow or decay.

Because of their inability to consistently increase or decrease and restrictions on the domain, exponential functions cannot have negative bases. Compound interest is a practical application for exponential functions that displays the restrictions on base values.

Explanation: A shrink of a function is a shrink on the vertical direction. It means that for a certain value of x, the new function will have a lower value, in the intervals where the function is positive, or a higher value, in those intervals where the function is negative.

When we multiply the parent function f(x)=bx f ( x ) = b x by –1, we get a reflection about the x-axis. When we multiply the input by –1, we get a reflection about the y-axis. For example, if we begin by graphing the parent function f(x)=2x f ( x ) = 2 x , we can then graph the two reflections alongside it.

Key Points If b>1 , the graph stretches with respect to the y -axis, or vertically. If b<1 , the graph shrinks with respect to the y -axis. In general, a horizontal stretch is given by the equation y=f(cx) y = f ( c x ) .

Shift the graph of f(x)=bx f ( x ) = b x left c units if c is positive and right c units if c is negative. Shift the graph of f(x)=bx f ( x ) = b x up d units if d is positive and down d units if d is negative.

Graphing an Exponential Function with a Vertical Shift constant k is what causes the vertical shift to occur. A vertical shift is when the graph of the function is moved up or down a fixed distance, k.

A horizontal shift is the result of adding a constant term to the function inside the parentheses. A positive term results in a shift to the left and a negative term in a shift to the right. A parent function is the simplest form of a particular type of function.

If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function.

Vertical shifts are outside changes that affect the output ( y- ) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( x- ) axis values and shift the function left or right.