1.2 Algebra and Functions

• Algebra uses symbols to represent numbers.


  True
• Algebra involves manipulating expressions, while functions relate inputs to outputs
• Linear equations are equations where the highest power of the variable is 1
• Quadratic equations have the general form ax^2 + bx + c = 0</latex>.

  True
• Functions relate inputs to outputs, with each input having only one output
• Functions are essential for modeling real-world relationships.

  True
• Inverse operations are used to isolate the variable in a linear equation.


  True
• Factorization involves rewriting the equation as a product of two linear factors.
• What is the quadratic formula used to solve for x in a quadratic equation?

  $x =$$\frac{ - b \pm \sqrt{b^{2} - 4ac}}{2a}$
• Algebra manipulates expressions, while functions relate inputs to outputs.


  True
• The quadratic formula is $x =$$( - b ± √(b^{2} - 4ac)) / 2a$, where $a, b, c$ are coefficients of the quadratic equation.
• What do functions describe the relationship between?

  Input and output variables
• What operations are used to isolate the variable in a linear equation?

  Inverse operations
• Steps to solve a quadratic equation using factorization:

  1️⃣ Rewrite the equation as a product of two linear factors

  2️⃣ Set each factor equal to zero

  3️⃣ Solve for x in each case
• What are simultaneous equations used to solve?

  Related algebraic quantities
• 1 + 2 = 3
• Steps for solving simultaneous equations using elimination

  1️⃣ Multiply equations to match coefficients

  2️⃣ Add or subtract equations

  3️⃣ Solve for the remaining variable
• Methods for solving quadratic equations

  1️⃣ Factorization

  2️⃣ Completing the square

  3️⃣ Quadratic formula
• There are three main methods to solve quadratic equations
• Solving quadratic equations is only useful in mathematics and not in physics or engineering.

  False
• Arrange the steps for solving simultaneous equations using the substitution method.

  1️⃣ Solve one equation for one variable

  2️⃣ Substitute that expression into the other equation

  3️⃣ Solve for the remaining variable
• If $f(x) =$$x^{2} +$$3x - 2$, then $f(3) =$$16$. This shows how the function maps the input x = 3</latex> to the output 16
• What is a composite function created from?

  The output of one function becoming the input of another
• Arrange the steps for finding the inverse of a function $f(x)$.


  1️⃣ Swap the $x$ and $y$ variables

  2️⃣ Solve the equation for $x$ in terms of $y$

  3️⃣ Rename the variables
• What effect does changing the slope of a linear function have on its graph?

  It affects the steepness
• A horizontal shift moves the function left or right
• Steps to transform a linear function `f(x) = mx + b` using transformations

  1️⃣ Change the slope `m` to affect steepness

  2️⃣ Change the y-intercept `b` to shift vertically
• The solution to the linear equation $3x - 7 =$$14$ is x = 7
• The method of completing the square involves manipulating the equation to the form (x - h)^2 = k
• In the example system \begin{cases} 2x +$y =$$7 \\ x - y =$ 2 \end{cases}, what is the value of y</latex> after substituting $x$?


  $y =$$1$
• Simultaneous equations are used to solve problems involving related quantities expressed algebraically
• Functions define relationships between input and output variables.


  True
• What is the general form of a quadratic equation?

  $ax^{2} +$$bx +$$c =$$0$
• What is the quadratic formula used to solve quadratic equations?

  $x =$$( - b ± √(b^{2} - 4ac)) / 2a$
• What are simultaneous equations used to solve?

  Problems involving related quantities
• The elimination method involves manipulating equations to cancel out one variable by adding or subtracting them.

  True
• The domain of a function is the set of all possible output values.

  False
• If $f(x) =$$x^{2}$ and g(x) = 2x + 1</latex>, then $f(g(x)) =$$4x^{2} +$$4x +$$1$. This shows how the composite function maps the input $x$ to the output $4x^{2} +$$4x +$$1$
• What are the main types of function transformations?

  Shifting, stretching/compressing, reflecting
• A vertical shift of a linear function changes its y-intercept but not its slope.


  True