Math 1010 Test 3 Review PDF

Chapter 5 and 6 Review Prof. Ibarra Campos Math 1010 Utah Valley University April 9, 2025 Non-comprehensive study guide for midterm 3:...

Chapter 5 and 6 Review Prof. Ibarra Campos Math 1010 Utah Valley University April 9, 2025 Non-comprehensive study guide for midterm 3: Chapters 5 and 6. Chapter 5 Chapter 5 includes: Exponential growth is discussed and exponential functions are developed using tables and graphs. Domain, range, and asymptotes are considered. Graphs are transformed and the corresponding algebraic functions are discussed. Exponential functions are manipulated algebraically through addition, subtraction, multiplication, and division. The rules of exponents are revisited and equivalent functions are written using these rules. Exponential equations are solved and reference back to finding intersection points on a graph of two functions. Composition of functions is explored and the inverse of an exponential function is investigated. 5.1 - Exponential parent graph y = bx. - Determine the pattern of exponential growth: Repeatedly multiplying an initial value by a constant ratio. Forming a J-shaped graph. If we want to know the values below the initial value, then we multiply the previous value by the reciprocal of the constant ratio. Remember that for any exponential term of the form b0 = 1, for a positive real number b. For instance, the value of the exponential 3x decreases in value after the initial value like: Figure 1: y = 3x For negative exponents, we can also just use the negative exponent rule of b−x = 1 bx. For instance, for 2x we have: 1  Figure 2: y = 2x - Define and graph exponential functions: Exponential functions are of the form y = a · bx , for an exponential growth rate/base b with initial value a. For exponential parent graphs of the form y = bx , they have a J-shaped graph when b > 1, and an horizontally inverted J-shaped when 0 < b < 1. Remember that the rate/base b must be strictly a real positive number, and b ̸= 0, 1. - Describe the asymptote of an exponential function: Exponential graphs of the form y = bx , have an horizontal asymptote of y = 0 (the x-axis). Remember that the asymptote can change when altering the parent function y = bx with a vertical shift. - Find the domain and range of an exponential function: For y = bx domain is all real numbers (−∞, ∞). While the range is all positive real numbers (0, ∞). Remember that the range can change when altering the parent function y = bx with a vertical shift. 5.2 - For the exponential function of the form y = bx , we can transform it with the form y = ab(x−h) + k: a: Represents an outside transformation of stretching, compressing, and/or vertical reflection across the x-axis. Meaning (x, y) → (x, ay). h: Represents an inside transformation of horizontal shift. Meaning (x, y) → (x + h, y). k: Represents an outside transformation of horizontal shift. Meaning (x, y) → (x, y + k). Which changes the y-intercept, and horizontal asymptote to k. Remember that for horizontal reflections across the y-axis, the equation becomes = ab−(x−h) + k, so (x, y) → (−x, y) 5.3 - Apply the product rule for exponents: bx · by = bx+y. x - Apply the quotient rule for exponents: bby = bx−y x x·r - Apply the power rule for exponents: (bx )y = bx·y. If ( ab y )r = ab y·r. And (bx ay )r = bx·r ay·r. - Explain the meaning of a zero exponent b0 = 1. - Explain the meaning of a negative exponent b−x = b1x. - Know how to simplify exponential expressions. - Use properties of exponents to write equivalent exponential functions in standard form: f (x) = 2x+3 → f (x) = 8 · 2x. 2 5.4 - Describe the meaning of solving exponential equations: The solution of f (x) = g(x) is the x-value of the intersection point(s) of the graph of f (x) and g(x). Exponential functions can only cross with each other one time. Therefore we must arrive to a unique solution. - Use the property of equality to solve exponential equations: If bx = by then x = y. 5.5 - Graph the inverse function of an exponential function by swapping the domain and range values, i.e., (x, y) → (y, x). Inverse graphs are reflections of each other along the y = x line. - The inverse function of an exponential function, is a logarithmic equation with the same base: f (x) = bx → f −1 (x) = logb (x). Where for all real numbers x, and positive real numbers a and b, b ̸= 1, if y = bx , then x = logb (y). - Know how to solve for the inverse of an exponential function with transformations. Chapter 6 Chapter 6 introduces: The logarithmic function as a function that solves for exponents as well as the inverse function of the exponential function. The graph of the logarithmic function, along with its domain, range and asymptotes, will be discussed. Previous knowledge of transformations will be applied to the logarithmic function both graphically and algebraically. The inverse function will be explored. Common and natural logarithms will be introduced along with the change of base formula. Rules for logarithms will be developed from exponential rules and used to simplify and expand loga- rithmic functions and expressions. Finally logarithmic equations will be solved algebraically with reference back to the graphical meaning of finding intersection points. 6.1 - Logarithmic parent graph y = logb (x) - Define logarithm: A logarithm is a quantity representing the power to which a fixed number (rate/base) must be raised to produce a given number. For all real numbers x, and positive real numbers a and b, b ̸= 1, if y = bx , then x = logb (y). Remember since b0 = 1 then logb (1) = 0. - Explain “logarithmic growth”: Unlike exponential growth that increases fast by multiplying by a constant each time, logarithmic growth increases very slowly. For example, the Figure 3 shows logarithmic growth with base 2. Notice that the value of the logarithm increases by only 1 as the value of the input (i.e., x) doubles each time. 3  Figure 3: y = log2 (x) - Define and graph logarithmic functions: Logarithmic functions have parent function of the form y = logb (x), they have a shape when b > 1, and an vertically inverted shaped when 0 < b < 1. Remember that the rate/base b must be strictly a real positive number, and b ̸= 0, 1. - Describe the asymptote of a logarithmic function: Logarithmic graphs of the form y = logb (x), have an vertical asymptote of x = 0 (the y-axis). Remember that the asymptote can change when altering the parent function with an horizontal shift. - Find the domain and range of a logarithmic function: For y = bx domain is all positive real numbers (0, ∞). While the range is all real numbers (−∞, ∞). Remember that the domain can change when altering the parent function with an horizontal shift. 6.2 - For the logarithmic parent function of the form y = logb (x), we can transform it with the form y = alogb (x − h) + k. - Know how each transformation affects the graph. Similarly to exponential functions (and any other function). - Remember that for horizontal reflections across the y-axis, the equation becomes = alogb (−(x − h)) + k, so (x, y) → (−x, y) 6.3 Graph the inverse function of logarithmic function by swapping the domain and range values, i.e., (x, y) → (y, x). Inverse graphs are reflections of each other along the y = x line. - The inverse function of a logarithmic function, is a exponential equation with the same base: f (x) = logb (x) → f −1 (x) = bx. Where for all real numbers x, and positive real numbers a and b, b ̸= 1, if y = logb (x), then x = by. - Know how to solve for the inverse of an exponential function with transformations. - Use compositions of function to show equivalency f (f −1 ))(x) = x for logarithmic and exponential functions. - Know the identity properties: blogb (x) = x and logb (bx ) = x. 6.4 - Apply the product rule for logarithms: logb (x · y) = logb (x) + logb (y). - Apply the quotient rule for logarithms: logb ( xy ) = logb (x) − logb (y). 4 - Apply the power rule for logarithms: logb (xr ) = r · logb (x). - Apply the change of base rule: For any real positive number m, m ̸= 1, then logb (x) = logm (x) logm (b). - Explain natural logarithms and common logarithms: The common base is when b = 10, so log10 (x) → log(x). And the natural logarithm is when b = e, so loge (x) → ln(x). - Know how to simplify logarithmic expressions. 6.5 - Describe the meaning of solving logarithmic equations: The solution of f (x) = g(x) is the x-value of the intersection point(s) of the graph of f (x) and g(x). Logarithmic functions can only cross with each other one time. Also, since they are one-to-one functions, we must arrive to a unique solution. - Know how to solve logarithmic equations using the rules for logarithms. - Solve exponential equations using logarithms and change of basis. - See attached example on how to solve an exponential equation with three different approaches: By using equality of bases (because it is possible for this specific example): Figure 4: Using equality of bases By using log8 : Figure 5: Using log8 By using the common base log: 5 Figure 6: Using common base log10 6