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--- math:pre-ap_calculus [2025/05/14 15:53] – [Chap.2 Function] root1
+++ math:pre-ap_calculus [2025/05/14 16:47] (current) – [Transformations of Functions] root1
@@ Line 4: / Line 4: @@
 ===== Chap.2 Function =====
+==== What is a Function? ====
 > Function Definition
@@ Line 12: / Line 14: @@
 {{url>https://www.desmos.com/calculator/pkb2vug0cf?embed 500,500 fullscreen | This is not a function}}
+=== Evaluating a Function ===
+\[
+\frac{f(x + h) - f(x)}{h}, \quad h \neq 0
+\]
+### Step 1: Compute \( f(x + h) \)
+\[
+f(x + h) = 2(x + h)^2 + 3(x + h) - 1
+\]
+\[
+= 2(x^2 + 2xh + h^2) + 3x + 3h - 1
+\]
+\[
+= 2x^2 + 4xh + 2h^2 + 3x + 3h - 1
+\]
+### Step 2: Compute \( f(x + h) - f(x) \)
+\[
+f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 + 3x + 3h - 1\right) - \left(2x^2 + 3x - 1\right)
+\]
+\[
+= 4xh + 2h^2 + 3h
+\]
+### Step 3: Form the Difference Quotient
+\[
+\frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 + 3h}{h}
+\]
+\[
+= \frac{h(4x + 2h + 3)}{h}
+\]
+\[
+= 4x + 2h + 3 \quad \text{(for \( h \neq 0 \))}
+\]
+=== The Domain of a Function ===
+> Recall that the //domain// of a function is the set of all inputs for the function.
+If the function is given by an algebraic expression and the domain is not stated explicitly, then by
+//convention the domain of the function is the domain of the algebraic expression—that
+is, the set of all real numbers for which the expression is defined as a real number.//
+=== Four Ways to Represent a Function ===
+  - verbally (by a description in words)
+  - algebraically (by an explicit formula)
+  - visually (by a graph)
+  - numerically (by a table of values)
+==== Graphs of Functions ====
+The most important way to visualize a function is through its graph. In this section
+we investigate in more detail the concept of graphing functions.
+=== Graphing Functions ===
+=== Graphing Piecewise Defined Functions ===
+\[
+f(x) =
+\begin{cases}
+x^2 & \text{if } x \leq 1, \\
+x + 1 & \text{if } x > 1.
+\end{cases}
+\]
+==== Increasing and Decreasing Functions; Average Rate of Change ====
+Functions are often used to model changing quantities. In this section we learn how
+to determine if a function is increasing or decreasing, and how to find the rate at
+which its values change as the variable changes.
+=== Increasing and Decreasing Functions ===
+**Increasing Function:**
+A function \( f \) is increasing on an interval \( I \) if
+\[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) \leq f(x_2). \]
+**Strictly Increasing Function:**
+A function \( f \) is strictly increasing on an interval \( I \) if
+\[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) < f(x_2). \]
+**Decreasing Function:**
+A function \( f \) is decreasing on an interval \( I \) if
+\[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) \geq f(x_2). \]
+**Strictly Decreasing Function:**
+A function \( f \) is strictly decreasing on an interval \( I \) if
+\[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) > f(x_2). \]
+=== Average Rate of Change ===
+Definition: For a function \( f(x) \), the average rate of change over the interval \([a, b]\) is given by:
+\[
+\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
+\]
+This formula calculates the change in the function's value (\( \Delta y \)) divided by the change in the input (\( \Delta x \)).
+==== Transformations of Functions ====
+In this section we study how certain transformations of a function affect its graph.
+This will give us a better understanding of how to graph functions. The transformations
+we study are shifting, reflecting, and stretching.
+=== Vertical Shifting ===
+Adding a constant to a function shifts its graph vertically: upward if the constant is
+positive and downward if it is negative.
+Use the graph \( f(x) = x^2 \) of to sketch the graph of each function.
+\[
+a.\quad  g(x) = x^2 + 3 \\
+b.\quad  h(x) = x^2 - 2
+\]
+{{url>https://www.desmos.com/calculator/uoacsjlmc5?embed 500,500}}
+=== Horizontal Shifting ===
+**Shift Right by \( h \) Units**
+\[
+y = f(x - h)
+\]
+**Example:** \( f(x) = x^2 \) shifted right by 2:
+\[
+y = (x - 2)^2
+\]
+**Shift Left by \( h \) Units**
+\[
+y = f(x + h)
+\]
+**Example:** \( f(x) = \sqrt{x} \) shifted left by 3:
+\[
+y = \sqrt{x + 3}
+\]
+**Key Rule**
+**\( -h \) inside \( f(x - h) \) → shifts right**
+**\( +h \) inside \( f(x + h) \) → shifts left**
+=== Reflecting Graphs ===
+To graph \( y = -f(x) \), reflect the graph of \( y = f(x) \) in the x-axis.
+To graph \( y = f(-x) \), reflect the graph of \( y = f(x) \) in the y-axis.
+=== Vertical Stretching and Shrinking ===
+To graph \( y = cf(x) \),
+If \( c > 1 \), stretch the graph of \( y = f(x) \) vertically by a factor of c.
+If \( 0 < c < 1 \), shrink the graph of \( y = f(x) \) vertically by a factor of c.
+=== Horizontal Stretching and Shrinking ===
+To graph \( y = f(cx) \),
+If \( c > 1 \), stretch the graph of \( y = f(x) \) vertically by a factor of 1/c.
+If \( 0 < c < 1 \), shrink the graph of \( y = f(x) \) vertically by a factor of 1/c.
+=== Even and Odd Functions ===
+If a function \( f \) satisfies \( f(x) = f(-x) \) for every number \( x \) in its domain, then \( f \) is called an even function.
+If \( f \) satisfies \( f(x) = -f(-x) \) for every number \( x \) in its domain, then \( f \) is called an odd function.
+==== Quadratic Functions; Maxima and Minima ====