1.2 Algebra and Functions

• What are the two types of solutions for algebraic equations and inequalities?

  Specific values and ranges
• To solve equations like $f(x) =$$g(x)$, you need to find the value(s) of $x$ for which both sides are equal
• Inequalities define a range of values that satisfy the condition.
• What does the domain of an algebraic function $f(x)$ represent?


  All possible input values
• The domain of $f(x) =$$\sqrt{x}$ includes all real numbers.


  False
• Match the graph feature with its description:

  Intercepts ↔️ Points where the graph crosses the axes

  Turning Points ↔️ Maximum or minimum points

  Asymptotes ↔️ Lines the graph approaches but never touches
• Steps to find the x-intercepts of $f(x) =$$x^{2} - 4x +$$3$


  1️⃣ Set $y =$$0$

  2️⃣ Solve $x^{2} - 4x +$$3 =$$0$

  3️⃣ Factor the equation $(x - 1)(x - 3) =$$0$

  4️⃣ Find the values $x =$$1$ and $x =$$3$
• What are the key features to identify when sketching algebraic functions?

  Intercepts and asymptotes
• Intercepts are where the graph crosses the x-axis and y-axis
• Asymptotes are lines that the graph approaches but never touches.
• Match the feature with its description for $f(x) =$$\frac{1}{x}$:


  X-intercept ↔️ None

  Y-intercept ↔️ (0, undefined)

  Asymptotes ↔️ x=0, y=0
• What are the asymptotes of the graph f(x) = \frac{1}{x}</latex>?

  x=0 and y=0
• The graph of $f(x) =$$\frac{1}{x}$ has no intercepts.
• In which quadrants does the graph of $f(x) =$$\frac{1}{x}$ lie?


  Quadrants I and III
• Understanding graph features helps in analyzing its behavior
• Transformations such as shifts, stretches, compressions, or reflections can affect the graph of a function.
• What does the transformation $f(x - a)$ represent in terms of the graph?


  Horizontal shift
• A vertical shift of $f(x)$ by $b$ units is represented by f(x) + b</latex>
• Asymptotes are lines that the graph approaches but never touches
• Intercepts are points where the graph crosses the axes.
• Match the transformation with its equation:

  Horizontal Shift ↔️ $f(x - a)$

  Vertical Shift ↔️ $f(x) +$$b$

  Horizontal Stretch ↔️ $f(\frac{x}{a})$

  Vertical Stretch ↔️ $a \cdot f(x)$
• Order the steps to transform $f(x) =$$x^{2}$ to $f(x - 3) =$$(x - 3)^{2}$


  1️⃣ Identify the type of transformation

  2️⃣ Substitute $x$ with $x - 3$

  3️⃣ Apply the substitution in the equation

  4️⃣ Simplify the equation if necessary
• What is the effect of $f(x - c)$ on the graph of $f(x)$?


  Horizontal shift
• The transformation $f(x + c)$ shifts the graph to the left if $c > 0$.
• The transformation $af(x)$ scales the graph vertically by a factor of a
• What is the effect of $- f(x)$ on the graph of $f(x)$?


  Vertical reflection
• What is the effect of multiplying the function by -1, i.e., $- f(x)$?


  Reflection across x-axis
• What is the effect of adding a constant to the input, e.g., $f(x + c)$?


  Horizontal shift left by c
• Vertical scaling involves multiplying the function by a constant, e.g., $af(x)$.
• Multiplying the input by a constant, e.g., $f(ax)$, results in horizontal scaling.
• What is the effect of adding a constant to the function, e.g., $f(x) +$$c$?


  Vertical shift up by c
• What is the effect of multiplying the input by a constant, e.g., $f(ax)$?


  Horizontal scaling
• Multiplying the input by -1, e.g., $f( - x)$, results in reflection across the y-axis.
• Order the key transformations of algebraic functions from simple to complex effects on the graph.

  1️⃣ Vertical shifts

  2️⃣ Horizontal shifts

  3️⃣ Vertical scaling

  4️⃣ Horizontal scaling

  5️⃣ Reflection
• What is the effect of multiplying the function by a constant, e.g., $af(x)$?


  Vertical scaling
• What is the effect of adding a constant to the function, e.g., $f(x) +$$c$?


  Vertical shift up by c
• Multiplying the input by a constant, e.g., $f(ax)$, results in horizontal scaling.
• Multiplying the input by -1, e.g., $f( - x)$, results in reflection across the y-axis.
• Match the transformation with its effect on the graph:

  $f(x) +$$c$ ↔️ Vertical shift up by c

  $f(x - c)$ ↔️ Horizontal shift right by c

  $- f(x)$ ↔️ Reflection across x-axis

  $f(ax)$ ↔️ Horizontal scaling
• What is the effect of adding a constant to the function, e.g., f(x) + c</latex>?

  Vertical shift up by c