# Digital SAT Math Prep Notes ## Overview This study guide is designed to help you prepare for the Digital SAT Math section. It includes key concepts, problem-solving strategies, and examples to reinforce understanding. The questions are weighted based on the College Board's content domains, ensuring accuracy and relevance. You'll encounter a mix of easy, medium, and hard questions, so practice thoroughly to build confidence. --- ## Key Concepts and Problem-Solving Strategies ### 1. **Linear Equations and Systems** Linear equations are foundational in SAT Math. They often appear in word problems, graphs, and systems of equations. #### **Example Problem: Cost of Renting a Backhoe** The cost of renting a backhoe for up to 10 days is $270 for the first day and $135 for each additional day. Write an equation for the total cost, $Y$, for $X$ days. - **Step 1:** Identify the fixed cost and the variable cost. - Fixed cost (first day): $270 - Variable cost (additional days): $135(X - 1)$ - **Step 2:** Write the equation: $$Y = 270 + 135(X - 1)$$ Simplify: $$Y = 135X + 135$$ Always account for the fixed cost separately when writing equations for scenarios with an initial fee. --- ### 2. **Exponential Growth** Exponential growth problems often involve geometric sequences or exponential functions. #### **Example Problem: Bacteria Growth** A culture of bacteria grows exponentially. On Day 1, the population is $2.5 \times 10^5$. On Day 2, it doubles to $5.0 \times 10^5$. Write a function to model the population, $F(n)$, after $n$ days. - **Step 1:** Recognize the growth factor is $2$ (population doubles each day). - **Step 2:** Write the function: $$F(n) = 2.5 \times 10^5 \cdot 2^{n-1}$$ - The $n-1$ accounts for Day 1 being the starting point. #### **Find the Day When Population Reaches $5.12 \times 10^8$** - Set $F(n) = 5.12 \times 10^8$: $$2.5 \times 10^5 \cdot 2^{n-1} = 5.12 \times 10^8$$ - Solve for $n$: $$2^{n-1} = \frac{5.12 \times 10^8}{2.5 \times 10^5} = 2048$$ $$n-1 = \log_2(2048) = 11$$ $$n = 12$$ The population reaches $5.12 \times 10^8$ on **Day 12**. --- ### 3. **Quadratic Equations** Quadratic equations appear in various forms, including standard form ($ax^2 + bx + c = 0$) and vertex form ($a(x-h)^2 + k$). #### **Example Problem: Discriminant and Number of Solutions** The quadratic equation $-x^2 + bx - 676 = 0$ has no real solutions. What is the greatest possible value of $b$? - **Step 1:** Use the discriminant formula: $$\Delta = b^2 - 4ac$$ - For no real solutions, $\Delta < 0$. - **Step 2:** Substitute $a = -1$, $c = -676$: $$\Delta = b^2 - 4(-1)(-676) = b^2 - 2704$$ - **Step 3:** Solve for $b$: $$b^2 < 2704 \implies b < \sqrt{2704} = 52$$ - The greatest integer value of $b$ is **51**. Forgetting to check the sign of $a$ when calculating the discriminant can lead to errors. --- ### 4. **Geometry and Trigonometry** Geometry questions often involve triangles, circles, and coordinate geometry. Trigonometry questions focus on sine, cosine, and tangent. #### **Example Problem: Right Triangle and Trigonometric Ratios** In a right triangle, $\sin R = \frac{15}{17}$. What is $\cos R$? - **Step 1:** Recall the Pythagorean identity: $$\sin^2 R + \cos^2 R = 1$$ - **Step 2:** Substitute $\sin R = \frac{15}{17}$: $$\left(\frac{15}{17}\right)^2 + \cos^2 R = 1$$ $$\frac{225}{289} + \cos^2 R = 1$$ $$\cos^2 R = \frac{64}{289}$$ $$\cos R = \frac{8}{17}$$ Use the Pythagorean theorem to verify side lengths when working with trigonometric ratios. --- ### 5. **Systems of Equations** Systems of equations often involve substitution or elimination methods. #### **Example Problem: Running and Biking** Morgan ran $R$ miles at 5 mph and biked $B$ miles at 10 mph. She ran and biked a total of 200 miles, and she biked for twice as many hours as she ran. How many miles did she bike? - **Step 1:** Write equations for total distance and time: $$R + B = 200$$ $$\frac{R}{5} = 2 \cdot \frac{B}{10}$$ - **Step 2:** Simplify the time equation: $$\frac{R}{5} = \frac{B}{5} \implies R = B$$ - **Step 3:** Substitute $R = B$ into the distance equation: $$B + B = 200 \implies B = 160$$ Morgan biked **160 miles**. --- ### 6. **Circle Geometry** Circle problems often involve arc lengths, sectors, and central angles. #### **Example Problem: Arc Length** In a circle, the minor arc $ADC$ has a length of $5\pi$, and the central angle $X = 100^\circ$. What is the length of the major arc $ABC$? - **Step 1:** Use the proportion of the circle's circumference: $$\frac{\text{Arc Length}}{\text{Circumference}} = \frac{\text{Central Angle}}{360^\circ}$$ - Circumference = $2\pi r$. - **Step 2:** Solve for the major arc: - Minor arc corresponds to $100^\circ$, so major arc corresponds to $360^\circ - 100^\circ = 260^\circ$. - Scale factor: $\frac{260}{100} = 2.6$. - Major arc length = $5\pi \cdot 2.6 = 13\pi$. The length of the major arc is **$13\pi$**. --- ### 7. **Exponents and Radicals** Simplify expressions involving exponents and radicals using properties of powers. #### **Example Problem: Simplify $\frac{\sqrt[5]{x^5}}{\sqrt[3]{x^4}}$** - **Step 1:** Rewrite radicals as fractional exponents: $$\frac{x^{5/5}}{x^{4/3}} = \frac{x^1}{x^{4/3}}$$ - **Step 2:** Subtract exponents: $$x^{1 - 4/3} = x^{-1/3}$$ When dividing exponents with the same base, subtract the powers. --- ## Practice Problems 1. A parabola has a vertex at $(9, -14)$ and intersects the x-axis at two points. Write the equation in vertex form and convert it to standard form. 2. Solve for $x$: $2x^2 - 5x - 3 = 0$ using the quadratic formula. 3. A circle has a radius of 7. Find the area of a sector with a central angle of $60^\circ$. --- ## Key Takeaways - **Linear equations**: Focus on slope-intercept form and systems of equations. - **Exponential growth**: Recognize geometric sequences and growth factors. - **Quadratics**: Use the discriminant to determine the number of solutions. - **Geometry**: Apply similarity, trigonometric ratios, and circle theorems. - **Exponents**: Simplify using fractional exponents and exponent rules. Practice consistently and review mistakes to strengthen your understanding of key concepts.