Midterm Review Guide Answer Keys (1-5)

# IM3 Midterm Review - Unit 1 (Equivalence) Name: Key ## 1. Write two equivalent expressions for each of the following. Your two expressions must be able to be simplified back into the original expression. - **a.** 15x * 10x + 5x - 1. 15x * 10x + 5x - 2. 8x + 7x - **b.** 12x * 6x * 4 * 2x...

# IM3 Midterm Review - Unit 1 (Equivalence) Name: Key ## 1. Write two equivalent expressions for each of the following. Your two expressions must be able to be simplified back into the original expression. - **a.** 15x * 10x + 5x - 1. 15x * 10x + 5x - 2. 8x + 7x - **b.** 12x * 6x * 4 * 2x * 5 - 1. 12x * 6x * 4 * 2x * 5 - 2. 4x³ * 3x⁶ - **c.** 7x / 3 * x + x + x + x + x + x + x + x - 1. 7x / 3 * x + x + x + x + x + x + x + x - 2. 3x + 4x - **d.** 16x³ - 28 / 4 * (4x -7) - 1. 16x³ - 28 / 4 * (4x - 7) - 2. 4x - 7 ## 2. Create a monomial and a trinomial. Then, multiply them together to get an equivalent sum. 3x(x² + 2x - 5) = 3x³ + 6x² - 15x ## 3. Create a binomial and a binomial. Then, multiply them together to get an equivalent sum. Show work. (2x + 3)(x - 4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12 ## 4. Create a binomial and a trinomial. Then, multiply them together to get an equivalent sum. Show work. (x + 2)(x² - 4x + 3) = x³ - 4x² + 3x + 2x² - 8x + 6 = x ³ - 2x² - 5x + 6 ## 5. Create 3 binomials. Then, multiply them together to get an equivalent sum. Show work. (x + 2)(x + 3)(x + 4) = (x + 2)(x² + 7x + 12) = x³ + 7x² + 12x + 2x² + 14x + 24 = x³ + 9x² + 26x + 24 ## 6. Which of the following are equivalent to 3 + x - (2x² + 5x - 1)? For each one, state if it is or is not equivalent and show/explain why or why not. 3 + x - 2x² - 5x + 1 = -2x² - 4x + 4 - **a)** -2x² - 4x + 4 - Equivalent, this is the simplified answer - **b)** -2x² - 4x + 2 - Not equivalent - should be a 4 - **c)** 2x² - 4x + 4 - Not equivalent - should be -2x² - **d)** -2(x² + 2x - 2) - Equivalent, when simplified # IM3 Midterm Review - Unit 2 (Quadratics) Name: Key ## 1. Which of the following are quadratic functions? Explain why or why not. - **f(x) = 2x(x² - 3x)** (Highest exponent is 3) - Not quadratic - **g(x) = 4x² + 11x - 3** (Highest exponent is 2) - Quadratic - **h(x) = 3(5x + 6)** (Highest exponent is 1) - Not quadratic ## 2. a. How many possible x-intercepts can a quadratic equation have? Sketch graphs to show this. - Two - One - None ## 2. b. Which part of the Quadratic Formula tells you how many x-intercepts your quadratic function will have? Explain how you can use this part to determine the number of x-intercepts. - **Discriminant (b² - 4ac)** - If the discriminant is **positive**, there are **two** x-intercepts. - If the discriminant is **zero**, there is **one** x-intercept (a double root). - If the discriminant is **negative**, there are **no** x-intercepts. ## 3. Find the x-intercepts by factoring and Zero Product Property. Show all work. - **a.** f(x) = 6x² + 12x - 6x(x + 2) = 0 - 6x = 0 x + 2 = 0 - (0, 0) (-2, 0) - **b.** h(x) = x² - 10x + 25 - (x - 5)(x - 5) = 0 - (x - 5)² = 0 - x - 5 = 0 - (5, 0) - **c.** g(x) = 3x² + 10x - 8 - 3x² + 12x - 2x - 8 = 0 - 3x(x + 4) - 2(x + 4) = 0 - (3x - 2)(x + 4) = 0 - 3x - 2 = 0 x + 4 = 0 - (2 / 3, 0) (-4, 0) - **d.** y = 4x² - 9 - (2x - 3)(2x + 3) = 0 - 2x - 3 = 0 2x + 3 = 0 - (3 / 2, 0) (-3 / 2, 0) ## 4. Find the y-intercept of each quadratic equation. Show all work. - **a.** f(x) = x² - 2x - 8 - (0)² - 2(0) - 8 - (0, -8) - **b.** g(x) = 3x² - 9x - 3(0)² - 9(0) - ( 0, 0) - **c.** y = 5x² + 4x - 7 - 5(0)² + 4(0) - 7 - (0, -7) # IM3 Midterm Review - Unit 3 (Vertex Form) Name: Key ## 1. a. Write the equation in vertex form of a parabola that has a vertex at (3, -1) - y=(x - 3)² - 1 ## 1. b. Take the parabola you wrote in part a and make it open down. - y = -(x - 3)² - 1 ## 1. c. Take the parabola you wrote in part a and make it vertically stretched. - y = 2(x - 3)² - 1 ## 2. Write the equation in vertex form of a parabola that opens up, is vertically compressed, and has a vertex at (-2, 5). - y= 1 / 2 (x + 2)² + 5 ## 3. What is the vertex of y = -x² - 5? State whether it opens up or down, is vertically stretched or compressed, if it shifts left/right (how much), and if it shifts up/down (how much). - Vertex: (0, -5) - Opens down - Vertically compressed - Vertical translation down 5 ## 4. Complete the square to put the following into vertex form. Then, state the vertex - **a.** f(x) = x² - 8x + 2 - x² - 8x + 16 = -2 + 16 - f(x) = (x - 4)² + 14 - Vertex: (4, 14) - **b.** g(x) = x² + 3x - 5 - x² + 3x + 2.25 = 5 + 2.25 - g(x) = (x + 1.5)² - 7.25 - Vertex: (-1.5, -7.25) ## 5. Create your own quadratic equation in standard form, y = ax² + bx + c, where the a-value is a 1. Then, complete the square to put it in vertex form and state the vertex. - y = x² + 6x + 10 - x² + 6x + 9 = - 10 + 9 - y = (x + 3)² + 1 - Vertex = (-3, 1) # IM3 Midterm Review - Unit 4 (Transformations) Name: Key ## 1. Name the function. Then, graph each parent graph by plotting 3 points and any asymptotes that exist. Finally, write the graphing form for each. - **a.** y = 2* - Exponential - y = a(x - h) + k - **b.** y = x³ - Cubic - y = a(x - h)³ + k - **c.** y = |x| - Absolute Value - y = a|x - h| + k - **d.** y = √x - Square Root/Radical - y = a√x - h + k - **e.** y = x² - Quadratic - y = a(x - h)² + k - **f.** y = 1/x - Rational - y = a / (x - h) + k ## 2. State the locator point and a-value. Then, write an equation in graphing form for each function. - **a.** Locator point (3, 2) a = 1 - y = (x - 3)³ + 2 - **b.** Locator point (-5, 5) a = none - y = x + 5 + 5 - **c.** Locator point (0, 4) a = 1 - y = √x + 4 - **d.** Locator point (0, 2) a = none - y = 2x + 2 - **e.** Locator point (4, -3) a = -1 - y = -1(x - 4)- 3 - **f.** Locator point (-5, 2) a = 2 - y = 2(x + 5)² + 2 - **g.** Locator point (6, -5) a = 3 - y = 3(x + 6)³ - 5 - **h.** Locator point (5, 4) a = none - y = (x - 5) + 4 - **i.** Locator point (7, -6) a = 4 - y = 4(x - 7) - 6 ## 3. Name the type of function. Then, describe the transformations applied to each function. This includes reflected over the x-axis, vertically stretched/compressed/neither, and translated right, left, up, or down (by how much). - **a.** y = -2(x + 4)² - 3 - Quadratic, vertical stretch, reflected over x-axis, horizontal translation left 4, vertical translation down 3 - **b.** y = 3√x - 5 - 3 - Square root, vertical stretch, horizontal translation right 5, vertical translation down 3 - **c.** y = 1 / (x + 1) + 6 - Rational, horizontal translation left 1, vertical translation up 6 - **d**. y = - |x - 7| - Absolute value, reflected over x-axis, vertical comperssion, horizontal translation right 7 - **e.** (x - 2)² + (y + 5)² = 16 - Circle, center is (2, -5), radius is 4, horizontal translation right 2, vertical translation down 5 - **f.** y = -2^x + 8 - Exponential, reflected over x-axis, vertical translation up 8 # IM3 Midterm Review - Unit 5 (Intersection) Name: Key ## 1. Graph the following by hand on the axes provided and find the intersection points. y = x - 3 y = (x - 2)² - 1 ### 1. a. Determine the intersection points visually from your sketch. - (2, -1) - (3, 0) ### 1. b. Now, solve algebraically. Show all work. - x - 3 = (x - 2)² - 1 - x - 2 = (x - 2)² - x - 2 = x² - 4x + 4 - 0 = x² - 5x + 6 - 0 = (x - 3)(x - 2) - x = 3 x = 2 - y = 3 - 3 = 0 (3, 0) - y = 2 - 3 = -1 (2, - 1) ## 2. a. Create an absolute value function that has been shifted left and down. Write the equation for a horizontal line that it would intersect twice. - y = |x + 3| - 2 - y = 4 ### 2. b. Graph both functions. ### 2. c. Determine the intersection points visually from your sketch. - (3, 4) - (-9, 4) ### 2. d. Now, solve algebraically. Show all work. - |x + 3| - 2 = 4 - |x + 3| = 6 - x + 3 = -6 x + 3 = 6 - x = -9 x = 3 # IM3 Midterm Review - Unit 5 (Intersection) Name: Key ## 3. a. Create a quadratic function that has been reflected over the x-axis, shifted right, and shifted down. Write the equation for a horizontal line that it would not intersect. - y = -(x - 2)² - 3 - y = 1 ## 3. b. Graph both functions. ## 3. d. Now, solve algebraically to show that the two functions do not intersect. Show all work. - -(x - 2)² - 3 = 1 - -(x - 2)² = 4 - √(x - 2)² = √4 - x - 2 = - 2 x - 2 = 2 ## 4. a. Graph the following by hand on the axes provided and find the intersection points. y = √x - 5 y = 7 - x ### 4. a. Determine the intersection points visually from your sketch. - (6, 1) ### 4. b. Now, solve algebraically. Show all work. - (√x - 5) = (7 - x)² - x - 5 = 49 - 14x + x² - 0 = x² - 15x + 54 - 0 = (x - 9)(x - 6) - x= 9 x= 6 - y = 7 - 6 = 1 (6, 1) ## 5. Find the intersection point between each set of linear functions. Show all work. - **a**. y = 3x - 2 - x - y = 4 - x - (3x - 2) = 4 - x - 3x + 2 = 4 - -2x = 2 - x = -1 - y = 3(-1)- 2 - y = -5 - **b**. 2(-2x + 2y = 6) - -4x + 2y = -18 - -2x + 2(-1) = 6 - -2x - 2 = 6 - -2x = 8 - x = -4 ## 6. a. Graph two lines that intersect at the point (3, 5). Then, write the equation of each line. - Line #1: y = 2x - 1 - Line #2: y = -x + 8 ## 6. b. Now, show algebraically that these lines do in fact intersect at the coordinate point (3, 5). Show all work. - 2x - 1 = -x + 8 - 3x = 9 - x = 3 - y = 2(3) - 1 - y = 5