Algebra Essentials Test
Algebra I • Comprehensive Assessment
Functions & Linear Modeling Test
Total Points: 100 pts
Name:
Date:
Period:
Score:
/ 100
1 Identify x- and y-Intercepts from a Graph
6 pts
-5-4-3-2-1 12345 54321 -1-2-3-4-5 x y
Identify the \(x\)-intercept and \(y\)-intercept of the graphed line shown. Write each key feature as a coordinate ordered pair \((x, y)\).
\(x\)-intercept:
\(y\)-intercept:
2 Rate of Change from an Equation
6 pts
Determine the rate of change (slope) of the linear function given by the equation below. Convert the equation to slope-intercept form and state \(m\).
\(6x - 4y = 20\)
Show Algebraic Work:
Rate of Change (\(m\)):
3 Parent Function Transformations
7 pts
Given the quadratic parent function \(f(x) = x^2\), determine all the effects and transformations that create the function \(g(x) = -3(x - 4)^2 + 7\).
Reflection & Vertical Stretch/Compression:
Horizontal & Vertical Translations:
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4 Rate of Change from a Table
7 pts
| \(x\) | \(y\) |
|---|
| \(-2\) | \(14\) |
| \(1\) | \(5\) |
| \(4\) | \(-4\) |
| \(7\) | \(-13\) |
Calculate the constant rate of change \(\frac{\Delta y}{\Delta x}\) for the linear function shown in the table. Show your ratio calculation.
Work & Final Answer:
5 Key Features in Real-World Context
7 pts
A city water storage reservoir is draining for maintenance according to the linear model \(W(t) = 450 - 25t\), where \(W(t)\) represents the water volume in thousands of gallons and \(t\) is time in hours.
A. Meaning of the \(y\)-intercept \((0, 450)\):
B. Meaning of rate of change \(-25\):
C. Find the \(t\)-intercept (\(W=0\)) and explain its real-world meaning:
6 Rate of Change from a Graph
6 pts
-5-4-3-2-1 12345 54321 -1-2-3-4-5 x y (-2, 4) (2, -4)
Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with the two marked coordinate points to calculate the exact rate of change of the graphed line.
Step-by-Step Calculation:
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7 Compare & Contrast Association vs. Causation
7 pts
Real-World Scenario: A coastal county collects data over two years and observes a strong positive correlation between daily beach ice cream sales and the daily number of lifeguard water rescues.
Does eating ice cream cause swimmers to need water rescue? Compare association and causation in this context, and identify an underlying lurking / confounding variable that explains this relationship.
Written Analysis & Explanation:
8 Graph Linear Function & Identify Key Features
7 pts
Graph the linear function \(f(x) = -\frac{3}{2}x + 3\) on the coordinate plane. Then identify each required key feature below.
-5-4-3-2-1 12345 54321 -1-2-3-4-5 x y
Slope (\(m\)):
\(y\)-intercept:
\(x\)-intercept:
9 Correlation Coefficient & Association Strength
7 pts
A statistician analyzes vehicle age in years (\(x\)) versus resale market value in thousands of dollars (\(y\)) and computes a linear correlation coefficient of \(r = -0.91\).
A. Direction & Strength Classification: Classify the direction (positive/negative) and strength (weak/moderate/strong).
B. Contextual Interpretation: Explain what \(r = -0.91\) specifically indicates about vehicle age and value.
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10 Graph Line Given Key Features (Intercept & Slope)
6 pts
Graph the linear function having a \(y\)-intercept at \((0, -2)\) and a slope \(m = \frac{4}{3}\). Write the function equation.
-5-4-3-2-1 12345 54321 -1-2-3-4-5 x y
Equation of line (\(y = mx + b\)):
Coordinates of a second point on the line:
11 Domain & Range of a Quadratic Function
7 pts
A quadratic function is defined by \(q(x) = -(x - 3)^2 + 5\). Its parabola opens downward with a maximum vertex at \((3, 5)\). Which statement best represents the domain and range of \(q(x)\)?
A. The domain is all real numbers, and the range is \(y \le 5\).
B. The domain is \(x \le 3\), and the range is all real numbers.
C. The domain is all real numbers, and the range is \(y \ge 5\).
D. The domain is \(x \ge 3\), and the range is \(y \le 5\).
Mathematical Justification / Vertex Reasoning:
12 Graph Line Given Key Features (Two Intercepts)
6 pts
Graph the linear function given an \(x\)-intercept at \((-4, 0)\) and a \(y\)-intercept at \((0, 2)\). Determine whether the function is increasing or decreasing.
-5-4-3-2-1 12345 54321 -1-2-3-4-5 x y
Calculated Slope (\(m\)):
Behavior (Circle one & justify):
[ ] Increasing [ ] Decreasing
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13 Solve Equation with Distributive Property
8 pts
Solve the linear equation for \(x\). Apply the distributive property, combine like terms, and isolate the variable.
\(4(2x - 3) + 7 = 3(x + 5) - 6\)
Step-by-Step Algebraic Work:
Final Solution:
\(x =\)
14 Determine Domain of a Linear Function
7 pts
A delivery courier charges fees modeled by the linear function \(C(m) = 2.50m + 5.00\), where \(C(m)\) is the total delivery cost in dollars and \(m\) is the delivery distance in miles. The company only makes deliveries between \(0\) and \(30\) miles. Which inequality best represents the domain of this function for this situation?
A. \(0 \le m \le 30\)
B. \(5.00 \le C(m) \le 80.00\)
C. All real numbers
D. \(m \ge 30\)
Explanation (Distinguish Domain Variable vs. Range Variable):
15 Solve Literal Equation for a Specified Variable
7 pts
The geometric formula for the area of a trapezoid is given below. Solve the literal equation for base \(b_1\). Isolate \(b_1\) completely.
\(A = \frac{1}{2}h(b_1 + b_2)\)
Show Inverse Operations:
Target Variable Form:
\(b_1 =\)
Academic Integrity Pledge:
I affirm that all work shown on this assessment is my own original work.
Signature:
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