Describe the shapes of each of the following types of function. For each one, state whether or not there are restrictions on the domain or range, and explain why.
Linear
Quadratic
Square Root
the domain of all polynomials (even linear or constant ones) is all reals
and you know that √x is only defined for x >= 0
Now google or look in your text for the graphs
1. Linear Functions:
A linear function represents a straight line on a graph and has the general form y = mx + b, where m is the slope of the line and b is the y-intercept. The graph of a linear function is always a straight line.
Restrictions on the Domain:
There are no specific restrictions on the domain for linear functions. They typically extend infinitely in both the positive and negative directions on the x-axis.
Restrictions on the Range:
Similarly, there are no specific restrictions on the range for linear functions. The range can extend infinitely in both the positive and negative directions on the y-axis.
2. Quadratic Functions:
A quadratic function represents a parabola on a graph and has the general form y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is U-shaped (upward or downward).
Restrictions on the Domain:
Quadratic functions have no restrictions on the domain, and they can have any real number as inputs.
Restrictions on the Range:
Quadratic functions have restrictions on the range depending on their shape. If the parabola opens upward, the range is limited to values greater than or equal to the minimum point (vertex) on the graph. If the parabola opens downward, the range is limited to values less than or equal to the maximum point (vertex).
3. Square Root Functions:
A square root function represents the square root of a variable on a graph and has the general form y = √x. The graph of a square root function is a half-parabola that opens towards the right.
Restrictions on the Domain:
Square root functions have restrictions on the domain since the square root of a negative number is undefined in the realm of real numbers. Therefore, the domain of a square root function is limited to values greater than or equal to zero (x ≥ 0).
Restrictions on the Range:
The range of a square root function is all nonnegative numbers (y ≥ 0) since the square root of any nonnegative number is a real number. It does not include negative numbers.