Algebra II

Algebra II in Grade 10 builds on the foundations of Algebra I by introducing more advanced mathematical concepts that connect equations, functions, and real-world problem solving. Instead of only solving basic equations, students now explore quadratic, exponential, and logarithmic functions, while also learning how to analyze data and apply algebra to practical contexts. This subject develops critical thinking by teaching learners to recognize patterns, manipulate algebraic expressions, and interpret graphs. By the end of the course, learners gain not only computational skills but also a deeper understanding of how algebra models relationships, supports scientific discovery, and strengthens problem-solving strategies in everyday life. Algebra II prepares students for higher-level mathematics and real-world decision-making.

🟢 Starter

• Explore how variables represent unknown values.
• Practice solving simple quadratic equations.
• Investigate how inequalities describe real-world limits.
• Explore how exponents simplify repeated multiplication.
• Reflect on how equations can model everyday situations.
• Research how graphs display relationships between numbers.
• Investigate how factoring breaks down quadratic expressions.
• Explore how absolute value equations represent distance.
• Practice using substitution in systems of equations.
• Explore how sequences organize repeating patterns.
• Research how algebra is applied in budgeting and finance.
• Investigate how coordinates locate points on a graph.
• Explore how slope shows the rate of change.
• Reflect on how formulas represent real-world rules.
• Practice rewriting expressions using distributive property.
• Explore how inequalities compare quantities.
• Investigate how square roots undo squaring.
• Practice solving word problems with equations.
• Explore how algebra connects to geometry.
• Reflect on why symbols make problem solving universal.

🟡 Practice

• Solve quadratic equations using the quadratic formula.
• Explore how exponential functions model population growth.
• Investigate how logarithms undo exponential growth.
• Analyze how systems of equations solve real-world problems.
• Explore how arithmetic and geometric sequences differ.
• Investigate how transformations shift or stretch graphs.
• Solve rational expressions and simplify fractions.
• Research how parabolas appear in physics.
• Explore how inequalities represent business constraints.
• Analyze how graphs compare linear and quadratic functions.
• Research how exponential growth applies to technology.
• Investigate how compound interest uses exponents.
• Solve quadratic equations by completing the square.
• Explore how graphing calculators assist problem solving.
• Analyze how tables and graphs show patterns.
• Research how logarithmic scales are used in science.
• Investigate how absolute value functions model distance.
• Solve real-world problems using proportions.
• Explore how data analysis relies on algebra.
• Analyze how quadratic regression predicts outcomes.

🔴 Challenge

• Debate whether mathematics is discovered or invented.
• Research how exponential decay models radioactive elements.
• Analyze how polynomials describe motion in physics.
• Investigate how matrices solve systems of equations.
• Propose how Algebra II skills prepare students for STEM careers.
• Debate whether all real-world problems can be solved algebraically.
• Research how complex numbers apply in engineering.
• Analyze how probability connects to algebraic functions.
• Investigate how algebra supports computer algorithms.
• Research how climate models use algebra.
• Debate whether abstract math has practical value.
• Investigate how trigonometry and algebra overlap.
• Analyze how optimization problems use inequalities.
• Research how encryption depends on algebraic concepts.
• Investigate how data modeling uses quadratic functions.
• Debate whether technology replaces or enhances algebra learning.
• Research how algebra underpins artificial intelligence.
• Analyze how graph theory supports networks.
• Investigate how algebra aids medical imaging.
• Propose how Algebra II prepares citizens for informed decision-making.

💡 Reflection Question
How can mastering Algebra II help you connect mathematical concepts to real-world challenges and prepare you for future opportunities?

1. Simplify the expression: (3x² – 4x + 8) – (x² – 2x + 1)
2. Solve the quadratic equation: 2x² + 5x – 3 = 0
3. Solve the system of equations:

y = -3x + 4 x + 4y = -6 

1. If f(x) = 2x² – 3x + 5, what is f(-1)?
2. Find the vertex of the parabola: defined by the equation y = 2x² – 8x + 3

1. Simplify the expression:
(3×2−4x+8)−(x2−2x+1)(3x² – 4x + 8) – (x² – 2x + 1)(3x2−4x+8)−(x2−2x+1)

Distribute the minus:
= 3×2−4x+8−x2+2x−13x² – 4x + 8 – x² + 2x – 13x2−4x+8−x2+2x−1
Combine like terms:
= (3×2−x2)+(−4x+2x)+(8−1)(3x² – x²) + (-4x + 2x) + (8 – 1)(3x2−x2)+(−4x+2x)+(8−1)
= 2×2−2x+72x² – 2x + 72x2−2x+7

✅ Final Answer: 2x² – 2x + 7

2. Solve the quadratic equation:
2×2+5x−3=02x² + 5x – 3 = 02x2+5x−3=0

Use quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Here a=2,b=5,c=−3a=2, b=5, c=-3a=2,b=5,c=−3.

Discriminant:
b2−4ac=25−4(2)(−3)=25+24=49b² – 4ac = 25 – 4(2)(-3) = 25 + 24 = 49b2−4ac=25−4(2)(−3)=25+24=49.

x=−5±494=−5±74x = \frac{-5 \pm \sqrt{49}}{4} = \frac{-5 \pm 7}{4}x=4−5±49​​=4−5±7​

Two solutions:

• x=−5+74=24=12x = \frac{-5 + 7}{4} = \frac{2}{4} = \tfrac{1}{2}x=4−5+7​=42​=21​

• x=−5−74=−124=−3x = \frac{-5 – 7}{4} = \frac{-12}{4} = -3x=4−5−7​=4−12​=−3

✅ Final Answer: x = ½ or x = -3

3. Solve the system of equations:

y=−3x+4y = -3x + 4 y=−3x+4 x+4y=−6x + 4y = -6x+4y=−6

Substitute y=−3x+4y = -3x + 4y=−3x+4 into second equation:

x+4(−3x+4)=−6x + 4(-3x + 4) = -6x+4(−3x+4)=−6 x−12x+16=−6x – 12x + 16 = -6x−12x+16=−6 −11x+16=−6-11x + 16 = -6−11x+16=−6 −11x=−22⇒x=2-11x = -22 \quad \Rightarrow \quad x = 2−11x=−22⇒x=2

Now substitute back:

y=−3(2)+4=−6+4=−2y = -3(2) + 4 = -6 + 4 = -2y=−3(2)+4=−6+4=−2

✅ Final Answer: (x, y) = (2, -2)

4. If f(x) = 2x² – 3x + 5, find f(-1):

f(−1)=2(−1)2−3(−1)+5f(-1) = 2(-1)² – 3(-1) + 5f(−1)=2(−1)2−3(−1)+5

= 2(1)+3+5=102(1) + 3 + 5 = 102(1)+3+5=10

✅ Final Answer: f(-1) = 10

5. Find the vertex of the parabola:
Equation: y=2×2−8x+3y = 2x² – 8x + 3y=2x2−8x+3

Vertex formula:

xv=−b2a,yv=f(xv)x_v = \frac{-b}{2a}, \quad y_v = f(x_v)xv​=2a−b​,yv​=f(xv​)

Here a=2,b=−8,c=3a=2, b=-8, c=3a=2,b=−8,c=3.

xv=−(−8)2(2)=84=2x_v = \frac{-(-8)}{2(2)} = \frac{8}{4} = 2xv​=2(2)−(−8)​=48​=2

Now plug x=2x=2x=2:

y=2(22)−8(2)+3=8−16+3=−5y = 2(2²) – 8(2) + 3 = 8 – 16 + 3 = -5y=2(22)−8(2)+3=8−16+3=−5

✅ Final Answer: Vertex = (2, -5)