1. What are the 7 categories of key characteristics of a parent function?
2. What are the key characteristics of a linear function, an absolute value function, and a reciprocal
function?
3. What are the differences between the two types of power functions, root functions, exponential functions,
and logarithmic functions?
4. How is this function transformed from its parent function? = 3√ + 4 − 3
5. What is the order that you perform multiple transformations?
1. The seven key categories of characteristics for a parent function are:
- Domain: The set of all possible x-values that the function can take.
- Range: The set of all possible y-values that the function can take.
- Symmetry: Whether the function is symmetric with respect to the y-axis, x-axis, or origin.
- Intercepts: The x-intercepts (where the graph crosses the x-axis) and y-intercepts (where the graph crosses the y-axis).
- Increasing and decreasing intervals: The intervals where the function is increasing or decreasing.
- Vertical asymptotes: The vertical lines that the graph approaches but never crosses.
- End behavior: Describes what happens to the graph as x approaches positive or negative infinity.
2. The key characteristics of a linear function include:
- Domain: All real numbers.
- Range: All real numbers.
- Slope: Determines the steepness of the line.
- Y-intercept: The point where the line crosses the y-axis.
- Linearity: The function has a constant rate of change.
The key characteristics of an absolute value function include:
- Domain: All real numbers.
- Range: All non-negative real numbers.
- Vertex: The lowest or highest point on the graph.
- Symmetry: The function is symmetric with respect to the vertical axis.
- Piecewise nature: The function's behavior changes at the vertex.
The key characteristics of a reciprocal function include:
- Domain: All real numbers except for zero.
- Range: All real numbers except for zero.
- Vertical asymptotes: Vertical lines where the function approaches but never crosses.
- Horizontal asymptotes: Horizontal lines where the function approaches as x approaches positive or negative infinity.
- Point of discontinuity: The function is undefined at x = 0.
3. Power functions: A power function has the form f(x) = ax^n, where a is the coefficient and n is a rational number. The key difference between the two types of power functions is whether n is a positive even integer (leading to even power functions) or a positive odd integer (leading to odd power functions).
Root functions: A root function has the form f(x) = √(x-a), where a is a constant. The differences between the two types of root functions depend on the power of the root. For example, a square root function (√x) and a cubic root function (∛x) have different behaviors.
Exponential functions: An exponential function has the form f(x) = a^x, where a is the base and x is the exponent. The key difference between exponential functions is the value of the base. For example, if a > 1, the function will grow exponentially, and if 0 < a < 1, the function will decay exponentially.
Logarithmic functions: A logarithmic function has the form f(x) = log_a(x), where a is the base. The differences between logarithmic functions depend on the value of the base. For example, if a > 1, the function will increase slowly, and if 0 < a < 1, the function will decrease slowly.
4. To understand how a function is transformed from its parent function, you need to look at the specific changes made to the parent function.
In the given example, the parent function is the cube root function: f(x) = ∛x.
The transformations applied to this parent function are:
- Vertical shift up 4 units: Adding 4 to the function shifts the entire graph up.
- Horizontal shift left 3 units: Subtracting 3 from x inside the function shifts the entire graph to the left.
- Vertical compression by a factor of 3: Multiplying the function by 3 compresses the graph vertically.
So, the transformed function becomes: g(x) = 3∛(x + 3) + 4
5. When performing multiple transformations on a function, the order in which you apply the transformations can affect the final result. The general order of transformations is:
1. Horizontal transformations: Includes shifts left or right.
2. Vertical transformations: Includes shifts up or down, reflections, and compressions or stretches.
3. Changes to the parent function: Includes any modifications to the function itself, such as scaling or adding constants.
By following this order, you can ensure that transformations are applied correctly and predict how the function will change. It's important to note that this is a general order, and specific situations may require adjustments.