Algebra Warm-Up Question Bank
Warm-Up Strand 1 TEKS A.2, A.3, A.5 & AR.3
Linear Systems & Functions
Name:
Date: Period:
Complete each daily warm-up prompt. Show all algebraic steps clearly in the provided workspace.
Day 1 • TEKS A.2.A / A.3.B (Rate of Change & Linear Models) Warm-Up 1.1
A mobile maker charges a flat initial equipment fee of \(\$45\) plus \(\$18\) per hour of on-site labor. Write a linear function \(C(h)\) representing the total charge for \(h\) hours, and calculate the total cost for a \(4.5\)-hour service call.
Function Model:
\(C(h) =\)
Evaluate for \(h = 4.5\):
Total Cost \(=\)
Day 2 • TEKS A.3.D / AR.3.A (Linear Inequalities) Warm-Up 1.2
Determine whether the point \((3, -2)\) is in the solution set of the linear inequality \(4x - 5y > 22\). Justify algebraically.
Substitution & Algebraic Verification:
Day 3 • TEKS A.5.A (Multi-Step Linear Equations) Warm-Up 1.3
Solve the linear equation for \(x\): \(3(2x - 5) + 4 = 5x - (x - 7)\).
Step-by-Step Solution:
Day 4 • TEKS A.5.C / AR.3.C (Systems of Linear Equations) Warm-Up 1.4
Solve the linear system using either substitution or elimination: \(\begin{cases} 2x + 3y = 12 \\ 4x - y = 10 \end{cases}\)
Work:
Ordered Pair Solution:
\((x, y) =\)
High School Mathematics • Texas Essential Knowledge and Skills (TEKS) Strand 1: Linear Functions • Page 1
Warm-Up Strand 2 TEKS A.6, A.7, A.8 & AR.3
Quadratic Functions & Modeling
Name:
Date: Period:
Analyze vertex, factored, and standard forms. Identify roots, domain, range, and extreme values.
Day 5 • TEKS A.6.B / AR.3.D (Key Features of Quadratics) Warm-Up 2.1
Given the quadratic function \(f(x) = -2(x - 3)^2 + 8\):
Vertex:
Axis of Sym:
Max / Min Value:
Range (Inequality):
Day 6 • TEKS A.8.A (Solving Quadratics by Factoring) Warm-Up 2.2
Solve by factoring over the real numbers: \(2x^2 + 7x - 15 = 0\).
Factored Form:
Solutions (\(x\)-intercepts):
Day 7 • TEKS A.7.A / A.8.B (Quadratic Formula & Radicals) Warm-Up 2.3
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the exact roots of \(x^2 - 6x + 4 = 0\). Express in simplest radical form.
Work & Simplification:
Day 8 • TEKS A.7.C / AR.3.E (Real-World Quadratic Application) Warm-Up 2.4
The height in feet of a golf ball is modeled by \(h(t) = -16t^2 + 64t\), where \(t\) is time in seconds. At what time \(t\) does the golf ball reach maximum height, and what is the maximum height attained?
Time to Max (\(t = -b/2a\)):
Maximum Height (\(h(t)\)):
High School Mathematics • Texas Essential Knowledge and Skills (TEKS) Strand 2: Quadratic Functions • Page 2
Warm-Up Strand 3 TEKS A.9 & AR.3.F-G, AR.4
Exponential Models & Growth
Name:
Date: Period:
Identify growth/decay factors, percentage rates of change, asymptotes, and exponential regression equations.
Day 9 • TEKS A.9.B / A.9.C (Growth vs. Decay Parameters) Warm-Up 3.1
An antique value is modeled by \(V(t) = 350(1.08)^t\), and a laptop value is modeled by \(L(t) = 1200(0.78)^t\), where \(t\) is years.
Antique \(V(t)\):
Type: [ Growth / Decay ]
Annual % Rate:
Initial Value:
Laptop \(L(t)\):
Type: [ Growth / Decay ]
Annual % Rate:
Initial Value:
Day 10 • TEKS A.9.D / AR.3.G (Formulating Exponential Functions from Data) Warm-Up 3.2
Write the exponential function in the form \(y = ab^x\) that passes through the table values:
Determine \(a\), \(b\), and write \(f(x)\):
\(f(x) =\)
Day 11 • TEKS A.9.A / AR.4.B (Asymptotes & Domain/Range) Warm-Up 3.3
For the exponential function \(g(x) = 5(2)^x - 4\), identify the equation of the horizontal asymptote, the domain, and the range.
Horizontal Asymptote:
Domain:
Range:
Day 12 • TEKS AR.4.A / AR.4.D (Model Comparison: Linear vs. Exponential) Warm-Up 3.4
Plan A adds \(\$150\) every month. Plan B deposits \(\$50\) and increases by \(10\%\) each month. Which plan yields more money at month \(1\) versus month \(36\)? Explain why exponential growth eventually surpasses linear growth.
Comparison & Explanation:
High School Mathematics • Texas Essential Knowledge and Skills (TEKS) Strand 3: Exponential Models • Page 3
Warm-Up Strand 4 TEKS A.10, A.11, A.12 & AR.2
Polynomials & Radical Operations
Name:
Date: Period:
Apply laws of exponents, polynomial arithmetic, factoring trinomials, and simplifying radicals.
Day 13 • TEKS A.11.B (Laws of Exponents & Radicals) Warm-Up 4.1
Simplify the expression so all exponents are positive: \(\frac{(3x^4 y^{-2})^3}{9x^{-2} y^5}\).
Step-by-Step Algebraic Simplification:
Day 14 • TEKS A.10.B / A.10.D (Polynomial Multiplication & Distribution) Warm-Up 4.2
Multiply and combine like terms to write in standard form: \((2x - 5)(3x^2 - 4x + 6)\).
Expanded Work:
Day 15 • TEKS A.10.E / A.10.F (Factoring Polynomials) Warm-Up 4.3
Factor each polynomial expression completely:
Part A: Difference of Squares
\(16x^2 - 81y^2\)
Part B: Trinomial Factoring
\(3x^2 - 11x - 20\)
Day 16 • TEKS A.11.A / AR.2.A (Radical Simplification & Operations) Warm-Up 4.4
Simplify the radical sum to simplest form: \(\sqrt{75x^3} - 2x\sqrt{27x} + \sqrt{48x^3}\), where \(x \ge 0\).
Radical Work & Final Simplified Expression:
High School Mathematics • Texas Essential Knowledge and Skills (TEKS) Strand 4: Operations & Polynomials • Page 4
Warm-Up Strand 5 TEKS AR.7.A - AR.7.E Standards Focus
Matrices & Algebraic Solutions
Name:
Date: Period:
Comprehensive warm-up coverage of Texas Algebraic Reasoning AR.7 matrix equations, operations, and model estimations.
Day 17 • TEKS AR.7.A (Matrix Equation Representation) Warm-Up 5.1
Represent the given system of 3 linear equations as a matrix equation of the form \(A \cdot X = B\): \[\begin{cases} 3x - 2y + z = 14 \\ x + 4y - 2z = -7 \\ 2x - y + 5z = 21 \end{cases}\]
Matrix Equation Formulation:
\(\begin{bmatrix} \phantom{000} & \phantom{000} & \phantom{000} \\ \phantom{000} & \phantom{000} & \phantom{000} \\ \phantom{000} & \phantom{000} & \phantom{000} \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \phantom{000} \\ \phantom{000} \\ \phantom{000} \end{bmatrix}\)
Day 18 • TEKS AR.7.B (Solving Linear Systems Using Matrices) Warm-Up 5.2
Find the inverse \(A^{-1}\) of the coefficient matrix \(A = \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix}\) using \(\det(A) = ad - bc\), then solve \(A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 18 \\ 7 \end{bmatrix}\).
Determinant & Inverse \(A^{-1}\):
Solution Vector \(X = A^{-1} B\):
Day 19 • TEKS AR.7.C (Matrix Operations & Polynomial Quotients) Warm-Up 5.3
Compute the matrix product \(M = 2\begin{bmatrix} 3 & -1 \\ 0 & 4 \end{bmatrix} + \begin{bmatrix} -4 & 5 \\ 2 & -3 \end{bmatrix}\), and determine the quotient of \(\frac{2x^3 - 5x^2 + 7x - 4}{x - 1}\).
Matrix Expression:
Polynomial Division Quotient:
Day 20 • TEKS AR.7.D / AR.7.E (Input Estimation & Model Solutions) Warm-Up 5.4
A biological population model is given by \(P(t) = \frac{800}{1 + 7(0.6)^t}\). Estimate the reasonable input value for time \(t\) that results in a population of \(P(t) = 500\), and interpret the validity of solutions in context.
Estimated Input (\(t\)):
Reasonableness & Context Interpretation:
High School Mathematics • Texas Essential Knowledge and Skills (TEKS) Strand 5: AR.7 Matrices & Advanced Algebraic Solutions • Page 5
