Algebra Ace SlidesALGEBRA ACE STAAR POWER-UP SESSION The Final Push 01 Simplifying Radicals The Game Plan: Find the largest **perfect square** that divides the number. Split the radical into two parts. Take the square root of the perfect square. Perfect Squares Cheat Sheet \(1^2 = 1\) \(2^2 = 4\) \(3^2 = 9\) \(4^2 = 16\) \(5^2 = 25\) \(6^2 = 36\) \(7^2 = 49\) \(8^2 = 64\) \(9^2 = 81\) PRACTICE: RADICALS Problem A \(\sqrt{72}\) Think: What is the biggest perfect square in 72? (Hint: 36) Problem B \(3\sqrt{20}\) Note: Multiply the outside number by the root you pull out! 02 Laws of Exponents PRODUCT \(x^a \cdot x^b = x^{a+b}\) Keep base, ADD powers POWER \((x^a)^b = x^{a \cdot b}\) Parentheses? MULTIPLY powers QUOTIENT \(\frac{x^a}{x^b} = x^{a-b}\) Stacking? SUBTRACT powers PRO TIP: Negative exponents cross the "fence" (fraction bar) to become positive! PRACTICE: EXPONENTS Problem C \((3x^4y^2)^3\) Don't forget the coefficient! \(3^3 = ?\) Problem D \(\frac{12a^5b^{-2}}{3a^2}\) Simplify numbers, subtract powers, move negatives. 03 Function Notation The Anatomy \(f(x) = y\) The number inside is your **Input** (\(x\)). The whole function is your **Output** (\(y\)). Evaluate vs. Solve Find \(f(5)\) Plug in 5 for every \(x\) and calculate. Find \(x\) if \(f(x) = 10\) Set the whole equation equal to 10 and solve for \(x\). PRACTICE: FUNCTIONS If \(f(x) = -2x^2 + 5x - 3\) Find \(f(-2)\) Watch your signs! \((-2)^2\) is positive 4. Find \(f(4)\) Square first, then multiply by -2. 04 Data & Modeling Rate of Change (Slope) \(m = \frac{y_2 - y_1}{x_2 - x_1}\) STAAR Keywords: "per", "each", "every", "rate". Initial Value (\(y\)-int) \(b\) Keywords: "start", "initial fee", "one-time", "fixed cost". The Modeling Strategy: 1 Find the slope (\(m\)) from any two points. 2 Use a point to find the start value (\(b\)). 3 Write your equation: \(y = mx + b\) THE FINAL CHALLENGE "A plumber charges an initial fee of **$50** plus **$75 per hour** of work. If the total bill was **$350**, how many hours did the plumber work?" Step 1: Equation \(y = 75x + 50\) Step 2: Plug in \(350 = 75x + 50\) Step 3: Solve \(x = ?\) VICTORY MINDSET TEST TIPS • Check your Reference Sheet! • Graph it if you're stuck. • Elimination works on MC. • Trust your work. YOU'VE GOT THIS. Complete your handout for final review.
Algebra Ace HandoutSTAAR POWER-UP ALGEBRA ACE ESSENTIALS Final Test Prep Assignment NAME: DATE: TOPIC 1: SIMPLIFYING RADICALS Perfect Square Refresher Strategy: \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\) Pull out the largest perfect square factor 1. Simplify: \(\sqrt{48}\) 2. Simplify: \(2\sqrt{75}\) TOPIC 2: LAWS OF EXPONENTS PRODUCT: \(x^a \cdot x^b = x^{a+b}\) POWER: \((x^a)^b = x^{ab}\) QUOTIENT: \(\frac{x^a}{x^b} = x^{a-b}\) 3. Simplify: \((4x^2y^5)^3\) 4. Simplify: \(\frac{24a^8b^3}{6a^2b^{-2}}\) TOPIC 3: FUNCTIONAL NOTATION Given the function: \(f(x) = -3x^2 + 4x + 10\) 5. Evaluate \(f(-3)\): 6. Evaluate \(f(2)\): 7. If \(g(x) = 5x - 12\), for what value of \(x\) does \(g(x) = 18\)? TOPIC 4: DATA MODELING 8. A car rental company charges a one-time insurance fee of $45 and $0.20 for every mile driven. Write a linear function \(C(m)\) to represent the total cost for driving \(m\) miles. Equation: C(m) = Cost for 150 miles: $ 9. The table below shows the linear relationship between the number of tickets sold (\(x\)) and the total revenue (\(y\)). Tickets (x)Revenue (y)10$15025$37540$600 Rate of Change (Revenue per ticket): Total Revenue for 100 Tickets: $ NO SHORTCUTS TO THE TOP Prepare. Practice. Prevail.
Algebra Ace KeyANSWER KEY ALGEBRA ACE ESSENTIALS Teacher Reference Guide TOPIC 1: SIMPLIFYING RADICALS Perfect Square Refresher 1 4 9 16 25 36 49 64 81 100 1. Simplify: \(\sqrt{48}\) \(\sqrt{16 \cdot 3} = 4\sqrt{3}\) 2. Simplify: \(2\sqrt{75}\) \(2 \cdot \sqrt{25 \cdot 3} = 2 \cdot 5\sqrt{3} = 10\sqrt{3}\) TOPIC 2: LAWS OF EXPONENTS 3. Simplify: \((4x^2y^5)^3\) \(4^3 \cdot x^{2 \cdot 3} \cdot y^{5 \cdot 3} = 64x^6y^{15}\) 4. Simplify: \(\dfrac{24a^8b^3}{6a^2b^{-2}}\) \(4a^{8-2}b^{3-(-2)} = 4a^6b^5\) TOPIC 3: FUNCTIONAL NOTATION The Function: \(f(x) = -3x^2 + 4x + 10\) 5. Evaluate \(f(-3)\): \(-3(-3)^2 + 4(-3) + 10\) \(-3(9) - 12 + 10 = -27 - 12 + 10 = -29\) 6. Evaluate \(f(2)\): \(-3(2)^2 + 4(2) + 10\) \(-3(4) + 8 + 10 = -12 + 8 + 10 = 6\) 7. Solve for \(x\) if \(g(x) = 5x - 12\) and \(g(x) = 18\): \(5x - 12 = 18 \rightarrow 5x = 30 \rightarrow x = 6\) TOPIC 4: DATA MODELING 8. Car Rental Scenario (Fee: $45, Rate: $0.20): Equation: C(m) = 0.20m + 45 150 miles: 0.20(150) + 45 = 30 + 45 = $75 9. Ticket Revenue Table Analysis: <table class="border-collapse border border-red-200 bg-white text-xs mb-4"><tbody><tr class="bg-red-50"><th class="p-2 border border-red-100 text-red-700">Tickets (x)</th><td class="p-2 border border-red-100">10</td><td class="p-2 border border-red-100">25</td><td class="p-2 border border-red-100">40</td></tr><tr><th class="p-2 border border-red-100 text-red-700">Revenue (y)</th><td class="p-2 border border-red-100">$150</td><td class="p-2 border border-red-100">$375</td><td class="p-2 border border-red-100">$600</td></tr></tbody></table> Rate of Change: \(\dfrac{375-150}{25-10} = \dfrac{225}{15} = $15\) per ticket 100 Tickets: \(15 \cdot 100 = $1,500\)
