Algebra 2 Past Paper PDF

# Algebra 2 ## Sec 3.1 Solving Quadratic Functions **EQ:** How can you use the graph of a quadratic equation to determine the number of real solutions of the equation? **Quadratic Equation in One Variable -** is an equation that can be written in the standard form *ax² + bx + c = 0* where *a, b, c...

# Algebra 2 ## Sec 3.1 Solving Quadratic Functions **EQ:** How can you use the graph of a quadratic equation to determine the number of real solutions of the equation? **Quadratic Equation in One Variable -** is an equation that can be written in the standard form *ax² + bx + c = 0* where *a, b, c* are real numbers and *a≠0*. **Root of an Equation -** is a solution of an equation. You can use various methods to solve quadratic equations. ### Solving Quadratic Equations * **By graphing:** Find the x-intercepts of the related function *y = ax² + bx + c*. * **Using square roots:** Write the equation in the form *u² = d*, where *u* is an algebraic expression, and solve by taking the square root of each side. * **By factoring:** Write the polynomial equation *ax² + bx + c = 0* in factored form and solve using the Zero-Product Property. ## EX 1: Solving Quadratic Equations by Graphing Solve each equation by graphing. **Use a calculator!** * *x² - x - 6 = 0* * *(x - 3)(x + 2) = 0* * *-2x² - 2 = 4x* * *-2x² - 4x - 2 = 0* ## EX 2: Solving Quadratic Equations Using Square Roots Solve each equation using square roots. * a. *4x² - 31 = 49* * *+31 + 31* * *4x² = 80* * *4 4* * *x² = 20* * *x = ±√20* * *x = ± 2√5* * b. *3x² + 9 = 0* * *-9 - 9* * *3x² = -9* * *3 3* * *x² = -3* * *x = ±√-3* * **No Solution,** the square of a real number can't be negative. ## EX 3: Solving a Quadratic Equation by Factoring Solve *x² - 4x = 45* by factoring. -45 -45 *x² - 4x - 45 = 0* *(x - 9)(x + 5) = 0* *x - 9 = 0* *x + 5 = 0* *x = 9* *x = -5* ## EX 4: Finding the Zeros of a Quadratic Function Find the zeros of *f(x) = 2x² - 11x + 12*. *2x² - 11x + 12 = 0* *(2x - 3)(x - 4) = 0* *2x - 3 = 0* *x - 4 = 0* *x = 3/2* *x = 4* ## EX 5: Solving a Multi-Step Problem A monthly teen magazine has *48,000* subscribers when it charges *20* dollars per annual subscription. For each *1* dollar increase in price, the magazine loses about *2000* subscribers. How much should the magazine charge to maximize annual revenue? What is the maximum annual revenue? * **Step 1:** Define the variables. * *x = price increase* * **Step 2:** Annual revenue = Number of Subscribers * Subscription price * *(dollars) = (# of people) * (dollars/person)* * *R(x) = (48,000 - 2000x) * (20 + x)* * *R(x) = (-2000x + 48,000)(20 + x)* * *R(x) = -2000(x - 24)(20 + x)* * **Step 3:** Identify the zeros and find their average. Then find how much subscription should cost. * Zeros are *24* and *-20*. * *24 + -20 / 2 = 2* * To maximize revenue, each subscription should cost *20 + 2 = 22* dollars. * **Step 4:** Find max annual revenue. * *R(2) = -2000(2 - 24)(2 + 20) = 968,000* dollars ## EX 6: Modeling a Dropped Object For science competition, students must design a container that prevents an egg from breaking when dropped from a height of *50* feet. Write a function that gives the height *(h)* (in feet) of the container after *(t)* seconds. How long does the container take to hit the ground? * Initial height is *50*, so the model is *h = -16t² + 50*. Find the zeros of the function. * *h = -16t² + 50* * *0 = -16t² + 50* * *-50 = -16t²* * *-50 / -16 = t²* * *√(50/16) = t* * *±1.8 ≈ t*. Since *t* has to be positive, only use *t = 1.8*. The container will fall for about *1.8* seconds before it hits the ground. Find and interpret *h(1) - h(1.5)*. * *h(1) = -16(1)² + 50 = -16 + 50 = 34* * *h(1.5) = -16(1.5)² + 50 = -16(2.25) + 50 = - 36 + 50 = 14* * *h(1) - h(1.5) = 34 - 14 = 20* * So the container fell *20* feet between *1* and *1.5* seconds. ## Algebra 2 ## Section 3.1 Worksheet 1. Solve the equation by graphing. * *x² + 3x + 2 = 0* * *2x = x² + 2* * *0 = x² - 9* 2. Solve the equation using square roots. * a. *(p - 4)² = 49* * b. *a² = 81* * c. *2(x + 2)² - 5 = 8* 3. Solve the equation by factoring. * a. *n² - 6n = 0* * b. *x² + 6x + 9 = 0* * c. *x² - 11x = -30* 4. Area of the rectangle is *36*. Find the value of *x*. * *x + 5 * x* ## Algebra 2 ## Section 3.1 Worksheet, Cont. 5. Solve the equation using any method. Explain your reasoning. * a. *u² = -9u* * b. *7(x - 4)² - 18 = 10* * c. *t² + 8t + 16 = 0* 6. Find the zeros of the function. * a. *f(x) = x² - 8x + 16* * b. *g(x) = x² + 11x* 7. You make a rectangular quilt that is *5* feet by *4* feet. You use the remaining *10* square feet of fabric to add a border of uniform width to the quilt. What is the width of the border? * *4 + 2x * x * 5 + 2x * x* ## Algebra 2 ## Sec 3.2 Complex Numbers **EQ:** What are the subsets of the set of complex numbers? **Complex Numbers** * **Real Numbers** * **Rational Numbers** * **Integers** * **Whole Numbers** * **Natural Numbers** * **Irrational Numbers** * **Imaginary Numbers** * The imaginary unit *i* is defined as *i = √-1*. **Imaginary Unit i -** defined as *i = √-1*. Note *i² = -1*. The imaginary unit *i* can be used to write the square root of any negative number. **The Square Root of a Negative Number** * **Property:** 1. If *r* is a positive real number, then *√-r = i√r*. 2. By the first property, it follows that *(i√r)² = -r*. ## EX 1: Finding Square Roots of Negative Numbers Find the square root of each number. * *√-25* * *√25 * -1* * *5i* * *√-72* * *√72 * √-1* * *√36 * √2 * i* * *6√2i* * *√-9* * *√9 * √-1* * *3i* * *√-54* * *2√54 * √-1* * *2√9 * √6 * i* * *2(3)√6i* * *6√6i*. ## EX 2: Equality of Two Complex Numbers Find the values of *x* and *y* that satisfy the equation *2x - 7i = 10 + yi*. * Set the real parts equal to each other, and set the imaginary parts equal to each other. * *2x = 10* * *x = 5* * *-7i = yi* * *-7 = y* So, *x = 5* & *y = -7*. ## PRACTICE: Equality of Two Complex Numbers Find the values of *x* and *y* that satisfy the equation. * *x + 3i = 9 - yi* * *9 + 4yi = -2x + 3i* ## EX 3: Adding and Subtracting Complex Numbers Add or subtract. Write the answer in standard form. * *(8 - i) + (5 + 4i)* * *(7 - 6i) - (3 - 6i)* * *13 - (2 + 7i) + 5i* ## EX 4: Solving a Real-Life Problem Electrical circuit components, such as resistors, inductors, and capacitors, all oppose the flow of current. This opposition is called resistance for resistors and reactance for inductors and capacitors. Each of these quantities is measured in ohms. The symbol used for ohms is Ω, the uppercase Greek letter omega. | Component and symbol | Resistance or reactance (in ohms) | Impedance (in ohms) | | ------------------------ | -------------------------------- | ------------------------ | | Resistor | *R* | *R* | | Inductor | *Li* | *Li* | | Capacitor | *-C* | *-C* | The table shows the relationship between a component's resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled. The impedance for a series circuit is *the sum of the impedances for the individual components*. Find the impedance of the circuit. * resistor = *5* ohms, so impedance = *5* ohms * inductor = *30* ohms, so impedance = *3i* ohms * capacitor = *4* ohms, so impedance = *-4i* ohms * **Impedance of circuit = 5 + 3i + (-4i) = 5 - i** ## EX 5: Multiplying Complex Numbers Multiply. Write the answer in standard form. * *4i(-6 + i)* * *(9 - 2i)(-4 + 7i)* ## PRACTICE: Multiplying Complex Numbers Perform the operation. Write the answer in standard form. * *(9 - i) + (-6 + 7i)* * *(3 + 7i)(8 - 2i)* * *(-3i)(10i)* * *(8 - i)²* * *(3 + i)(5 - i)* ## EX 6: Solving Quadratic Equations Solve *a) x² + 4 = 0* and *b) 2x² - 11 = -47*. ## EX 7: Finding Zeros of a Quadratic Function Find the zeros of *f(x) = 4x² + 20*. ## PRACTICE: Finding the Zeros of a Quadratic Function Find the zeros of the function. * *f(x) = x² + 7* * *f(x) = -x² - 4* * *f(x) = 9x² + 1* ## Algebra 2 ## Section 3.2 Worksheet 1. Which number does not belong with the other three? Explain your reasoning. * *3 + 0i* * *2 + 5i* * *√3 + 6i* * *0 - 7i* 2. Find the square root of the number. * a. *√-36* * b. *√-49* * c. *√-32* * d. *√-24* 3. Find the values of *x* and *y* that satisfy the equation. * a. *4x + 2i = 8 + yi* * b. *-12x + yi = 60 - 13i* * c. *2x - yi = 14 + 12i* 4. Add or subtract. Write the answer in standard form. * a. *(9 + 5i) + (11 + 2i)* * b. *7 - (3 + 4i) + 6i* * c. *(16 - 9i) - (2 - 9i)* 5. Write the expression as a complex number in standard form: *√-9 + √-36*. 6. Solve the equation. Check your solutions. * a. *x² + 9 = 0* * b. *x² + 49 = 0* * c. *x² - 4 = -11* 7. Find the zeros of the function. * a. *f(x) = 3x² + 6* * b. *h(x) = 2x² + 72* * c. *m(x) = x² - 27* ## Algebra 2 ## Sec 3.3 Completing the Square **EQ:** How can you complete the square for a quadratic expression? ## EX 1: Solving a Quadratic Equation Using Square Roots Solve *x² - 16x + 64 = 100* using square roots. ## PRACTICE: Solving a Quadratic Equation Using Square Roots Solve the equation using square roots. Check your solution(s). * *x² + 4x + 4 = 36* * *x² - 6x + 9 = 1* * *x² - 22x + 121 = 81* ## EX 2: Making a Perfect Square Trinomial Find the value of *c* that makes *x² + 14x + c* a perfect square trinomial. Then write the expression as the square of a binomial. ## PRACTICE: Making a Perfect Square Trinomial Find the value of *c* that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. * *x² + 8x + c* * *x² - 2x + c* * *x² - 9x + c* ## EX 3: Solving ax² + bx + c = 0 when a = 1 Solve *x² - 10x + 7 = 0* by completing the square. ## EX 4: Solving ax² + bx + c = 0 when a ≠ 1 Solve *3x² + 12x + 15 = 0* by completing the square. ## Algebra 2 ## PRACTICE: Solving by Completing the Square Solve the equation by completing the square. * *x² - 4x + 8 = 0* * *6x(x + 2) = -42* * *4x² + 32x = -68* ## EX 5: Writing a Quadratic Function in Vertex Form Write *y = x² - 12x + 18* in vertex form. Then identify the vertex. ## PRACTICE: Writing the Function in Vertex Form * *y = x² - 8x + 18* * *y = x² + 6x + 4* * *y = x² - 2x - 6* ## EX 6: Modeling with Mathematics The height *y* (in feet) of a baseball *t* seconds after it is hit can be modeled by the function *y = -16t² + 96t + 3*. Find the maximum height of the baseball. How long does the ball take to hit the ground? ## Algebra 2 ## Section 3.3 Worksheet 1. Solve each equation using square roots. Check your solution(s). * a. *x² - 8x + 16 = 25* * b. *r² - 10r + 25 = 1* * c. *y² - 24y + 144 = -100* 2. Find the value of *c* that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. * a. *x² + 10x + c* * b. *w² + 13w + c* * d. *s² - 26s + c* 3. Solve the equation by completing the square. * a. *x² + 6x + 3 = 0* * b. *t² - 8t - 5 = 0* * c. *6r² + 6r + 12 = 0* 4. Find the value of *c*. Then write an expression represented by the diagram. * *x * 2 * x² * 2x * 2 2x * c* * *x * 6 * x * x² * 6x * 6 6x * c* * *x * 8 * x * x² * 8x * 8 8x * c* 5. Describe and correct the error in finding the value of *c* that makes the expression a perfect square trinomial. * *x * x² + 30x + c * x² + 30x + 30 * x² + 30x + 15* ## Algebra 2 ## Sec 3.4 Using the Quadratic Formula **EQ:** How can you derive a general formula for solving a quadratic equation? **Quadratic Formula -** is a formula that gives you the **solutions** of any quadratic equation. The formula is: * *x = (-b ± √(b² - 4ac)) / 2a* ## EX 1: Solving an Equation with Two Real Solutions Solve *x² + 3x = 5* using the Quadratic Formula. ## PRACTICE: Solving an Equation with Two Real Solutions Solve the equation using the Quadratic Formula. * *x² - 6x + 4 = 0* * *2x² + 4 = -7x* * *5x² = x + 8* ## EX 2: Solving an Equation with One Real Solution Solve *25x² - 8x = 12x - 4* using the Quadratic Formula. ## EX 3: Solving an Equation with Imaginary Solutions Solve *-x² + 4x = 13* using the Quadratic Formula. ## PRACTICE: Solving Equations Using the Quadratic Formula Solve the equation using the Quadratic Formula. * *x² + 8x + 41 = 0* * *-9x² = 30x + 25* * *5x - 7x² = 3x + 4* ## Algebra 2 ## Discriminant **Discriminant -** is the expression *b² - 4ac* of the associated equation *ax² + bx + c = 0*. You can analyze the discriminant to determine the number and type of solutions of the quadratic equation. | Value of discriminant | Number and type of solutions | | ----------------------- | ------------------------------- | | *b² - 4ac > 0* | Two real solutions | | *b² - 4ac = 0* | One real solution | | *b² - 4ac < 0* | Two imaginary solutions | ## EX 4: Analyzing the Discriminant Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. * *a. x² - 6x + 10 = 0* * *b. x² - 6x + 9 = 0* * *c. x² - 6x + 8 = 0* ## PRACTICE: Analyzing the Discriminant Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. * *4x² + 8x + 4 = 0* * *x² + x - 1 = 0* * *-5x² + 16 = -10x* ## EX 5: Writing and Equation Find a possible pair of integer values for *a* and *c* so that the equation *ax² - 4x + c = 0* has one real solution. Then write the equation. ## PRACTICE: Writing an Equation Find a possible pair of integer values for *a* and *c* so that the equation *ax² + 3x + c = 0* has two real solutions. Then write the equation. ## Methods for Solving Quadratic Equations | Method | When to Use | | ------------------- | --------------------------------------------------------------------- | | Graphing | Use when approximate solutions are adequate. | | Using square roots | Use when solving an equation that can be written in the form *u² = d*, where *u* is an algebraic expression. | | Factoring | Use when a quadratic equation can be factored easily. | | Completing the square | Can be used for any quadratic equation *ax² + bx + c = 0* but is simplest to apply when *a = 1* and *b* is an even number. | | Quadratic Formula | Can be used for any quadratic equation. | ## Algebra 2 ## EX 6: Modeling Real-Life Problems The function *h = -16t² + v₀t + h₀* is used to model the height *h* of a dropped object. For an object that is launched or thrown, an extra term *v₀t* must be added to the model to account for the object's initial vertical velocity *v₀* (in feet per second). Recall that *h* is the height (in feet), *t* is the time in motion (in seconds), and *h₀* is the initial height (in feet). * *h = -16t² + h₀* Object is dropped. * *h = -16t² + v₀t + h₀* Object is launched or thrown. As shown below, the value of *v₀* can be positive, negative, or zero depending on whether the object is launched upward, downward, or parallel to the ground. ## EX 6: Modeling a Launched Object A juggler tosses a ball into the air. The ball leaves the juggler's hand *4* feet above the ground and has an initial vertical velocity of *30* feet per second. The juggler catches the ball when it falls back to a height of *3* feet. How long is the ball in the air? ## Algebra 2 ## Sec 3.5 Solving Nonlinear Systems **EQ:** How can you solve a **nonlinear system of equations**? **System of Nonlinear Equations -** a system of equations where at least one of the equations is nonlinear. ## EX 1: Solving a Nonlinear System by Graphing Solve the system by graphing. * *y = x² - 2x - 1* * *y = -2x - 1* ## EX 2: Solving a Nonlinear System by Substitution Solve the system by substitution. * *x² + x - y = -1* * *x + y = 4* ## EX 3: Solving a Nonlinear System by Elimination Solve the system by elimination. * *2x² - 5x - y = -2* * *x² + 2x + y = 0* ## PRACTICE: Solving Nonlinear Systems Solve the system using any method. Explain your choice of method. * *y = -x² + 4* * *y = -4x + 8* * *x² + 3x + y = 0* * *-2x + y = -5* * *2x² + 4x - y = -2* * *x² + y = 2* ## EX 4: Solving a Nonlinear System by Substitution Solve the system by substitution. * *x² + y² = 10* * *y = -3x + 10* ## PRACTICE: Solving Nonlinear Systems Solve the system. * *x² + y² = 16* * *y = -x + 4* * *x² + y² = 4* * *y = x + 4* * *x² + y² = 1* * *y = x + 1* ## Algebra 2 ## Solving Equations by Graphing **Step 1:** To solve the equation *f(x) = g(x)*, write a system of two equations, *y = f(x)* and *y = g(x)*. **Step 2:** Graph the system of equations *y = f(x)* and *y = g(x)*. The *x*-value of each solution of the system is a solution of the equation *f(x) = g(x)*. ## EX 5: Solving Quadratic Equations by Graphing Solve *(a) 3x² + 5x - 1 = -x² + 2x + 1* and *(b) (x - 1.5)² + 2.25 = 2x(x + 1.5)* by graphing. ## Algebra 2 ## Section 3.4 Worksheet, Cont. 5. Use the discriminant to match each quadratic equation with the correct graph of the related function. Explain your reasoning. * *x² - 6x + 25 = 0* * *2x² - 20x + 50 = 0* * *3x² + 6x - 9 = 0* * *5x² - 10x - 35 = 0* 6. Describe and correct the error in solving the equation. * *x² + 10x + 74 = 0* ## Algebra 2 ## Sec 3.6 Quadratic Inequalities **EQ:** How can you solve a **quadratic inequality**? **Quadratic Inequality in Two Variables -** can be written in one of the following forms, where *a, b, c* are real numbers and *a ≠ 0*. * *y < ax² + bx + c* * *y > ax² + bx + c* * *y ≤ ax² + bx + c* * *y ≥ ax² + bx + c* ## Graphing a Quadratic Inequality in Two Variables To graph a quadratic inequality in one of the forms above, follow these steps. * **Step 1:** Graph the parabola with the equation *y = ax² + bx + c*. Make the parabola dashed for inequalities with < or > and solid for inequalities with ≤ or ≥. * **Step 2:** Test a point *(x, y)* inside the parabola to determine whether the point is a solution of the inequality. * **Step 3:** Shade the region inside the parabola if the point from *Step 2* is a solution. Shade the region outside the parabola if it is not a solution. ## EX 1: Graphing a Quadratic Inequality in Two Variables Graph *y < -x² - 2x - 1*. ## EX 2: Using a Quadratic Inequality in Real Life A manila rope used for rappelling down a cliff can safely support a weight *W* (in pounds) provided *W ≤ 1480d²* where *d* is the diameter (in inches) of the rope. Graph the inequality and interpret the solution. ## EX 3: Graphing a system of Quadratic Inequalities Graph the system of quadratic inequalities. * *y < -x² + 3* * *y ≥ x² + 2x - 3* ## PRACTICE: Graphing Systems of Inequalities * Graph the inequality *y ≥ x² + 2x - 8*. * Graph the system of inequalities of *y ≤ -x²* and *y > x² - 3*. ## Quadratic Inequality in One Variable Can be written in one of the following forms, where *a, b, c* are real numbers and *a ≠ 0*. * *ax² + bx + c < 0* * *ax² + bx + c > 0* * *ax² + bx + c ≤ 0* * *ax² + bx + c ≥ 0* You can solve quadratic inequalities using algebraic methods or graphs. ## EX 4: Solving a Quadratic Inequality Algebraically Solve *x² - 3x - 4 < 0* algebraically. ## EX 5: Solving a Quadratic Inequality by Graphing Solve *3x - x - 5 ≥ 0* by graphing. ## EX 6: Modeling with Mathematics A rectangular parking lot must have a perimeter of *440* feet and an area of at least *8000* square feet. Describe the possible lengths of the parking lot. ## Algebra 2 ## Section 3.6 Worksheet 1. Match the inequality with its graph. * *y ≤ x² + 4x + 3* * *y > -x² + 4x - 3* * *y < x² - 4x + 3* * *y ≥ x² + 4x + 3* 2. Graph the inequality. * *a. y < -x²* * *b. y ≥ 4x²* * *c. y < x² + 5* 3. Describe and correct the error in graphing *y ≥ x² + 2*. 4. Graph the system of quadratic inequalities. * *y ≥ 2x²* * *y < -x² + 1* * *y ≥ 2x² - 3x - 6* * *y ≤ x² + 7x + 6* 5. Solve the inequality algebraically. * *a. 4x² < 25* * *b. x² + 10x + 9 < 0* 6. The graph shows a system of quadratic inequalities. * **a**. Identify two solutions of the system. * **b**. Are the points *(1, -2)* and *(5, 6)* solutions of the system? Explain. * **c**. Is it possible to change the inequality symbol(s) so that one, but not both, of the points in part *(b)* is a solution of the system? Explain. ## Algebra 2 ## Sec 3.5 Solving Nonlinear Systems **EQ:** How can you solve a **nonlinear system of equations**? **System of Nonlinear Equations -** a system of equations where at least one of the equations is nonlinear. ## EX 1: Solving a Nonlinear System by Graphing Solve the system by graphing. * *y = x² - 2x - 1* * *y = -2x - 1* ## EX 2: Solving a Nonlinear System by Substitution Solve the system by substitution. * *x² + x - y = -1* * *x + y = 4* ## EX 3: Solving a Nonlinear System by Elimination Solve the system by elimination. * *2x² - 5x - y = -2* * *x² + 2x + y = 0* ## PRACTICE: Solving Nonlinear Systems Solve the system using any method. Explain your choice of method. * *y = -x² + 4* * *y = -4x + 8* * *x² + 3x + y = 0* * *-2x + y = -5* * *2x² + 4x - y = -2* * *x² + y = 2* ## EX 4: Solving a Nonlinear System by Substitution Solve the system by substitution. * *x² + y² = 10* * *y = -3x + 10* ## PRACTICE: Solving Nonlinear Systems Solve the system. * *x² + y² = 16* * *y = -x + 4* * *x² + y² = 4* * *y = x + 4* * *x² + y² = 1* * *y = x + 1* ## Algebra 2 ## Section 3.5 Worksheet, Cont. 5. Solve each system by elimination. * *2r² - 3x - y = -5* * *-x + y = 5* * *-3x² + 2x - 5 = y* * *-x + 2 = -y* 6. Solve the system using any method. Explain your choice of method. *y = x² - 1* * *-y = 2x² + 1* * *-2x + 10 + y = x²* * *y = 10* ## Algebra 2 ## Sec 3.6 Quadratic Inequalities **EQ:** How can you solve a **quadratic inequality**? **Quadratic Inequality in Two Variables -** can be written in one of the following forms, where *a, b, c* are real numbers and *a ≠ 0*. * *y < ax² + bx + c* * *y > ax² + bx + c* * *y ≤ ax² + bx + c* * *y ≥ ax² + bx + c* ## Graphing a Quadratic Inequality in Two Variables To