Domain and Range Review Slides Day 01
ALGEBRA ACE
EOC Vocabulary Countdown
Domain
Range
Term 01
Domain
The set of all possible input values (\(x\)-values) for which the function is defined.
Visual Sample
{(1, 5), (2, 10), (3, 15)}
Domain: {1, 2, 3}
EOC Tip
Think left to right . On a graph, find the furthest left point and furthest right point.
x-axis = Domain
Term 02
Range
The set of all possible output values (\(y\)-values) resulting from the function.
Visual Sample
{(1, 5), (2, 10), (3, 15)}
Range: {5, 10, 15}
EOC Tip
Think bottom to top . On a graph, find the lowest point and the highest point.
y-axis = Range
PRACTICE
A function is represented by the set of ordered pairs below:
{(-4, 2), (-2, 5), (0, 8), (2, 11)}
What is the range of this function?
A. {-4, -2, 0, 2}
B. {2, 5, 8, 11}
C. {-4, 2, -2, 5}
D. {0, 2, 4, 6}
Correct Answer
B
The range is the set of \(y\)-values in the ordered pairs.
Slope and y-intercept Review Slides Day 02
Linear Logic
EOC Vocabulary Countdown
Slope
y-intercept
Term 01
Slope
The rate of change between any two points on a line. Often denoted as \(m\).
Key Formula
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Rise over Run
EOC Tip
Look for keywords like per , each , or every in word problems. These always signal the slope!
Term 02
y-intercept
The \(y\)-coordinate of a point where a graph crosses the \(y\)-axis . Occurs when \(x = 0\).
Visual Sample
\(y = 2x + \mathbf{5}\)
y-intercept: (0, 5)
EOC Tip
This is the starting value or initial amount in a real-world scenario. It's the "flat fee" or "base price."
Practice
A technician charges a one-time service fee of $50 plus $35 per hour of work. Which equation represents the total cost, \(y\), for \(x\) hours of work?
A. \(y = 50x + 35\)
B. \(y = 35x + 50\)
C. \(y = 85x\)
D. \(y = 50x\)
Correct Answer
B
The "per hour" value (35) is the slope, and the "one-time fee" (50) is the y-intercept.
Zeros and x-intercepts Review Slides Day 03
Crossing Axes
EOC Vocabulary Countdown
Zero
x-intercept
Term 01
Zero
The input value (\(x\)) that makes the function equal to zero (\(f(x) = 0\)).
Table View
The zero is 2
EOC Tip
"Zero" and "x-intercept" are often used interchangeably on the EOC. If you see one, think of the other!
Term 02
x-intercept
The point where a graph crosses the \(x\)-axis . The \(y\)-coordinate is always 0.
Coordinate Form
\( (x, 0) \)
Example: (4, 0)
Pro Discovery
To find the x-intercept from an equation, plug in 0 for \(y\) and solve for \(x\).
Practice
What is the zero of the linear function \(f(x) = \frac{1}{2}x - 6\)?
A. -6
B. 3
C. 6
D. 12
Correct Answer
D
Set \(f(x) = 0\):
Parent Patterns Review Slides Day 04
Parent Patterns
EOC Vocabulary Countdown
Parent Function
Transformation
Term 01
Parent Function
The simplest function in a family that preserves the shape of the entire family.
The "Big Two"
Linear
\(y = x\)
Quadratic
\(y = x^2\)
EOC Tip
Memorize these graphs! Linear is a straight diagonal line; Quadratic is a perfect U-shape (parabola) through the origin.
Term 02
Transformation
A change to the size, shape, or position of a parent function graph.
Common Types
\(f(x) + k\): Shift Up
\(f(x) - k\): Shift Down
\(-f(x)\): Reflect over x-axis
Vertex Trick
If you add a number at the end, the graph moves vertically . If you multiply by a negative, the graph flips .
Practice
How does the graph of \(g(x) = x^2 - 7\) compare to the parent function \(f(x) = x^2\)?
A. The graph is shifted up 7 units.
B. The graph is shifted down 7 units.
C. The graph is shifted left 7 units.
D. The graph is shifted right 7 units.
Correct Answer
B
Subtracting 7 at the end of the function causes a vertical shift down by 7 units.
Systems Showdown Review Slides Day 05
Systems Showdown
EOC Vocabulary Countdown
System of Equations
Solution
Term 01
System of Equations
A set of two or more equations with the same variables.
Typical Setup
\( \begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases} \)
EOC Tip
A system can have one solution (crossing lines), no solution (parallel lines), or infinite solutions (same line).
Term 02
Solution
The point (\(x, y\)) that makes both equations true. It is where the lines intersect .
Intersection Point
\( (x, y) \)
Think: Where do they cross?
Calculator Hack
Use the Intersection feature on your graphing calculator (2nd + TRACE + 5) to find the answer instantly!
Practice
A system of equations is graphed. One line has a slope of 2 and passes through (0, 0). The other line has a slope of -1 and passes through (0, 3). At what point do they intersect?
A. (0, 3)
B. (1, 2)
C. (2, 1)
D. (3, 0)
Correct Answer
B
The equations are \(y = 2x\) and \(y = -x + 3\).
Power Play Review Slides Day 06
Power Play
EOC Vocabulary Countdown
Exponent Laws
Polynomials
Term 01
Exponent Laws
Rules used to simplify expressions involving powers with the same base.
The Core Three
\(x^a \cdot x^b\) \(x^{a+b}\)
\(x^a / x^b\) \(x^{a-b}\)
\((x^a)^b\) \(x^{a \cdot b}\)
EOC Tip
Don't forget: any number (except 0) to the zero power equals 1 .
\(x^0 = 1\)
Term 02
Polynomial
An expression consisting of variables and coefficients, using addition, subtraction, and multiplication.
Anatomy
\( \mathbf{5}x^{\mathbf{3}} - 2x + 1 \)
Degree: 3 (Highest exponent)
Leading Coeff: 5
Pro Discovery
Polynomials are classified by their number of terms: 1 term (Monomial), 2 terms (Binomial), 3 terms (Trinomial).
Practice
Which expression is equivalent to \( \frac{(x^3)^4}{x^2} \)?
A. \(x^5\)
B. \(x^6\)
C. \(x^{10}\)
D. \(x^{14}\)
Correct Answer
C
Step 1: Power of a Power \(\rightarrow (x^3)^4 = x^{3 \cdot 4} = x^{12}\)
Quadratic Curves Review Slides Day 07
Quadratic Curves
EOC Vocabulary Countdown
Vertex
Axis of Symmetry
Term 01
Vertex
The highest (maximum) or lowest (minimum) point on a parabola.
Visual Sample
\(y = (x-2)^2 + \mathbf{3}\)
Vertex: (2, 3)
EOC Tip
If \(a\) is positive , it's a minimum (opens up). If \(a\) is negative , it's a maximum (opens down).
Term 02
Axis of Symmetry
A vertical line that divides the parabola into two matching halves.
Key Formula
\(x = \frac{-b}{2a}\)
It's always \(x = \dots\)
EOC Tip
The Axis of Symmetry always passes through the \(x\)-coordinate of the vertex. If the vertex is \((h, k)\), the axis is \(x = h\).
Practice
The vertex of a quadratic function is \((3, -4)\). What is the equation of the axis of symmetry?
A. \(y = -4\)
B. \(x = 3\)
C. \(x = -3\)
D. \(y = 3\)
Correct Answer
B
The axis of symmetry is always the \(x\)-coordinate of the vertex. Since the vertex is (3, -4), the axis is \(x = 3\).
Solving for Zero Review Slides Day 08
Solving for Zero
EOC Vocabulary Countdown
Roots / Zeros
Solutions
Term 01
Roots & Zeros
The \(x\)-values where the quadratic function crosses the \(x\)-axis (\(y = 0\)).
Key Concept
On the EOC, these 4 words mean the same thing :
Roots
Zeros
X-Ints
Solutions
EOC Tip
Think of "ROXS" :
Term 02
Solutions
The specific values of \(x\) that satisfy the quadratic equation \(ax^2 + bx + c = 0\).
Factored Sample
\( (x - 4)(x + 2) = 0 \)
Solutions: \(x = 4\), \(x = -2\)
Pro Discovery
In factored form, the solutions are the opposite of the numbers inside the parentheses.
Practice
What are the zeros of the function \( f(x) = (x + 5)(x - 8) \)?
A. {5, 8}
B. {-5, -8}
C. {-5, 8}
D. {5, -8}
Correct Answer
C
Set each factor to zero:
Exponential Trends Review Slides Day 09
Exponential Trends
EOC Vocabulary Countdown
Growth
Decay
Term 01
Growth
A function where the quantity increases by a constant percentage rate over time.
The Rule
\(y = a(b)^x\)
Growth if \(b > 1\)
EOC Tip
Keywords include doubles , triples , or appreciates . The graph curves steeply upward .
Term 02
Decay
A function where the quantity decreases by a constant percentage rate over time.
The Rule
\(y = a(b)^x\)
Decay if \(0 < b < 1\)
EOC Tip
Keywords include halves , depreciates , or loses value . The graph curves downward toward the x-axis.
Practice
The value of a car is modeled by the function \( V(t) = 25,000(0.85)^t \). Which statement is true?
A. The car's value is increasing by 85% each year.
B. The car's value is decreasing by 85% each year.
C. The car's value is increasing by 15% each year.
D. The car's value is decreasing by 15% each year.
Correct Answer
D
The base (0.85) is less than 1, so it is decay.
Final Features Review Slides Day 10
Final Features
EOC Vocabulary Countdown
Asymptote
Correlation Coefficient
Term 01
Asymptote
A line that a graph approaches but never quite touches or crosses.
Visual Sample
\(y = 3(2)^x + \mathbf{5}\)
Asymptote: \(y = 5\)
EOC Tip
For exponential functions \(y = ab^x + c\), the asymptote is always the value of \(c\).
Term 02
Correlation Coefficient
A number (\(r\)) between -1 and 1 that measures the strength and direction of a linear relationship.
Strength Scale
1 or -1 = Perfect Linear
Near 0 = Very Weak / No Link
Positive = Line goes up
Negative = Line goes down
Calculator Hack
To see \(r\) on your calculator, you must turn DiagnosticOn in the Catalog (2nd + 0) before running a Linear Regression.
Practice
A set of data has a correlation coefficient of \( \mathbf{r = -0.92} \). Which statement best describes the relationship between the variables?
A. Strong positive linear relationship.
B. Strong negative linear relationship.
C. Weak positive linear relationship.
D. Weak negative linear relationship.
Correct Answer
B
The coefficient is negative (line goes down) and the value 0.92 is very close to 1, making it a strong relationship.
