Time for the Algebra 2 Midterm!
Multiple Choice Question 10: Analyze a function IB Style Question 2: Analyze functions
-Functions are specific relations. -All functions are relations. -In a function, each input has exactly one output. -There can be multiple inputs for an output, there can't be multiple outputs for a input. -You can find out if a graph is a function by using the vertical line test (where you draw a vertical line in the middle of the graph, if the line passes through two points or more, it is not a function). -Linear, piecewise, step, and absolute value are all functions. -Continuous functions have an infinite set of points. -Discrete functions have a finite set of points.
Multiple Choice Question 6: Factor polynomial expressions. Part 2
-Rational Zero Theorem: -If you can't use anything else, the helpful but long rational zero theorem can help you out. -Ex. Shown in the picture -Degree of polynomial = how many zeros you should find.
Multiple Choice Question 3: Simplify rational expressions
-To simplify, remove the GCF from both the numerator and denominator. -May require factoring -Ex: Shown in picture -To divide, make sure to multiply the reciprocal of the second fraction to the first one. Then solve like shown in the picture.
Multiple Choice Question 20: Identify an appropriate function for a real-world scenario
Practice for this, because there's nothing really to say.
Multiple Choice Question 8: Simplify radical expressions
-A root is the inverse of an exponent. -Ex: x^3 is the inverse of 3√(x) -> 3√(x) = 64 -> x=4. -a^n = b where a is an nth root of b -what number times n equals your answer -Principle Root -equations must use ± -expressions only consider the principle root -when there's more than one real root and n is even, the non-negative root is called the principle root. -Ex: √(25) = 5 -√(25) = -5 ±√(25) = ±5 -When being asked to find a fraction exponent, it is the same thing as taking the root of something. -Ex: 27^1/3 = 3√(27) or 3 (-16)^1/2 = √(-16) or 4i - b^x/y = y√(b^x) = y√(b)^x -Ex 1: 27^2/3 = 3√(27)^2 = 3^2 or 9 -Ex 2: (-16)^3/2 = √(-16)^3 = (4i)^3 or 64i -If you find an even root of an even power and the result is a variable raised to an odd power, you need to include the absolute value symbols (||) around the variable because you need a positive value. -Ex: 4√(y)^4 -> (y^4)^1/4 -> (simplify (4/1)*(1/4)) y^1 = |y|. -Ex: 6√64(x^2-3)^18 = (64(x^2-3)^18)^1/6 -Distribute the ^1/6 to 64 and (x^2-3)^18: (64^1/6(x^2-3)^18)^1/6 -> ((2^6)^1/6 (x^2-3)^3 -> 2 |(x^2-3)^3| -Rules for Simplifying: -no negative exponents. -must have positive integers. -not a complex fraction (fraction within a fraction)
Multiple Choice Question 16: Find the average rate of change over an interval of a function
-Linear functions all have a constant gradient (IB for slope), but some functions don't have a constant rate of change. -When that happens we can calculate the average rate of change over a specified interval. -change in y over change in x (∆y/∆x). -"x" is the independent variable, "y" is the dependent variable. -Once you figured out what x and y is, put the subtraction of the y values over the subtraction of the x values. Ex: -change in elevation of Sun/change in time -(84º-6º)/(11:00 AM-7:00 AM) -78º/4 hours = 19.5 degrees per hour
Multiple Choice Question 21: Identify an equation of a polynomial given a graph
-Polynomials has its x's raised to powers. -Powers have to be whole and positive. -A degree of a polynomial is the greatest exponent of its variable. -The leading coefficient is the coefficient of the term with the highest degree. -Ex: 3x^2-2x^4-7+x^3 -Leading Coefficient: -2 -Degree: 4 -Location Principle (LP): if you want to find the solutions, the LP tells you to look between the negative and positive values. -Ex: x: -2, -1, 0, 1, 2 f(x): 35, -2, -5, -4, 19 -Look at the values between -2 and -1 and 1 and 2, because the signs change in between those inputs. -Maximum and minimum points (extrema): -Can have absolute and relative extrema. -Extrema is where the graph turns. -Relative maximum: highest point in a specific area on the graph. -Relative minimum: lowest point in a specific area on the graph. -Absolute maximum: highest point overall on the graph. -Absolute minimum: lowest point overall on the graph. -To find the smallest degree = number of turning points on the graph + 1. -This means that if you take the degree minus 1, you get the minimum number of points there are.
Multiple Choice Question 22: Solve a radical equation
-Steps to solving radical equations: -Isolate radical on one side of the equation. -Raise each side of equation to the power equal to the index of the radical to eliminate it. -Solve the resulting polynomial. -Check your answer. -Radical equations include radical expressions -which can be solved by raising each side of equation to a power. -results may be a number that doesn't satisfy the original equation, known as extraneous solutions. -Ex. Shown in picture. -If the solution ends up being extraneous, you can write "null set" or the symbol for null set, which is Ø
Multiple Choice Question 14: Rewrite an expression by completing the square Multiple Choice Question 18: Identify transforming values of a quadratic equation given the graph IB Style Question 1: Analyze and rewrite the form of quadratic functions
6-Completing the Square: -Manipulate equation so "c" is on the right side of the equation and the "a" = 1. -Ex: Solve x^2+10x-11 by completing the square -Add 11 to both sides: x^2+10x =11 -Take 1/2 of b and square your results (b/2)^2 - (10/2)^2 -> (5)^2 = 25 -Add result from last step to each side of the equation. -Add 25 to both sides: x^2+10x+25=11+25 = x^2+10x+25=36 -Rewrite the left side of the equation as a perfect square (x + b/2)^2 -Take the solution of b/2 (in this case: 5) and plug it in: (x+5)^2 = 36 -Solve using square root property. -Square root both sides of the equation: √(x+5)^2 = √(36) -Simplify and subtract 5 from both sides: x+5 = ±6 -> x=-5±6 -Solve: x=-5+6=1 or x=-5-6=-11 -If there is a value for "a", before completing the square divide that value to all 3 parts of the quadratic. -Ex: Solve 2x^2+4x-1=0 by completing the square: -Divide 2 to all 3 parts: (2/2)x^2+(4/2)x-1/2 = x^2+2x-1/2 -Add 1/2 from both sides: x^2+2x =1/2 -(b/2)^2 -> (2/2)^2 = 1, then add 1 to both sides -> x^2+2x+1=3/2 -Solve using the square root property: (x+1)^2= 3/2 -> √(x+1)^2=√(3/2) -Multiply √(3/2) by √(2)/√(2): x+1=√(3/2)*√(2)/√(2) -> x+1=±√(6)/2 -Subtract 1 from both sides: x=-1±√(6)/2 -Discriminant: expression radical sign, b^2-4ac, in the quadratic formula. -If b^2-4ac > 0 and a perfect square: 2 rational roots. -If b^2-4ac > 0 but not a perfect square: 2 irrational roots. -If b^2-4ac = 0: 1 rational root -If b^2-4ac < 0: 2 complex roots -Ex: Find type and solutions for 2x^2+3x-7=0 without solving the equation. - b^2-4ac -> (3)^2-4(2)(-7) - 9+56 = 65 > 0 -> 2 real irrational roots. -Vertex Form (makes the quadratic easy to graph): y=a(x-h)^2 + k -Written in terms of transforming values. -Direction of opening: upward - a>0, downward - a<0 -Vertex: (h,k) -Axis of Symmetry (AoS): a = h -Ex: Write x^2-10x+32 in vertex form -Plug in the numbers to find (b/2)^2 -> (-10/2)^2 = 25 -By adding 25 to x^2-10x, you must subtract 25, because it is basically like adding 0: (x^2-10x+25)+32-25 -Then turn x^2-10x into (x-5)^2: y = (x-5)^2 + 7 -> Vertex: (5,7) -If there is an "a" value, you will have to distribute it out of the quadratic. -Ex: Write y=-4x^2+16x-11 -> (factored -4) y=-4(x^2-4x )-11 - (b/2)^2 -> (-4/2)^2 = 4. -Add 4 to (x^2+4x ) and multiply that same number (4) to the number in front of the parentheses (-4), which gets -16. It is like you subtracted -16, so add 16 to -11: y=-4(x^2+4x+4)-11+16 = y=-4(x^2+4x+4)+5 -> y=-4(x-2)^2 + 5 -> Vertex: (2,5) -We can work backwards too. -Ex: Write an equation for a parabola with the vertex (1,3) passing through (-2, -15). -Plug in the vertex into vertex form: y=a(x-1)^2+3 -To find "a", plug in the known coordinate into the equation, -2 into the x's and -15 into the y: -15=a((-2)-1)^2+3 -Solve for "a": -15=a(-3)^2+3 -> -15=9a+3 -> -18=9a -> a=-2. -Plug the "a" value back into the vertex form equation to get your answer: y=-2(x-1)^2+3 -Radical functions may be graphed using transformations on the parent functions. - f(x) = a√b(x-h) + k -a: vertical stretch (multiplier of the y-values) -b: horizontal stretch (divisor of the x-values) -h: horizontal shift (subtracts from x-values) -k: vertical shift (adds to the y-values) -Outside of the radical = affects the y-values, does what you see to the y-values. -Inside the radical = affects the x-values, does the opposite of what you see to the x-values. -For square root functions: -Its parent function coordinates are -(0,0), (1,1), (4,2) -For cube root functions: -Its parent function coordinates are -(-1,-1), (0,0), (1,1) -To graph a radical, first look at whether it is a cube root function or a square root function. Then, grab the proper parent coordinates and make the changes to the y-values. Then, make the changes to the x-values. Finally, graph them and find the domain and range. -Ex. Look in your notes for a good example. -Tip: The domain and range for cube root functions are always the same: -Domain: (−∞, ∞) -Range: (−∞, ∞)
Multiple Choice Question 2: Perform operations on functions
-Addition: (f + g) (x) = f(x) + g(x) -Ex: Let f(x)=3x^2+7x and g(x)=2x^2-x-1 -(3x^2+7x) + (2x^2-x-1) -Just simplify: =5x^2+6x-1 -Subtraction: (f - g) (x) = f(x) - g(x) -Ex: Let f(x)=3x^2+7x and g(x)=2x^2-x-1 -(3x^2+7x) - (2x^2-x-1) -Distribute the negative to the second value: (3x^2+7x) -2x^2+x+1. -Simplify: =x^2+8x+1 -Multiplication: (f * g) (x) = f(x) * g(x) -Ex: Let f(x)=3x^2-2x+1 and g(x)=x-4 -(3x^2-2x+1)(x-4) -FOIL: 3x^3-12x^2-2x^2+8x+x-4 -Combine like terms (simplify): 3x^3-14x^2+9x-4 -Division: (f/g) (x) = f(x)/g(x), g(x) ≠ 0 -Ex: Let f(x)=3x^2-2x+1 and g(x)=x-4 -(3x^2-2x+1)/(x-4), x ≠ 4 -Note: to find the domain restriction (the "x ≠ 4" in this problem) when dividing functions, set the denominator equal to 0 and solve (Ex: x-4 = 0 -> x=4 -> x ≠ 4). -Ex 2: Let f(x)=x^2-16 and g(x)=x-4 -f(x)/g(x) = (x^2-16)/(x-4), x ≠ 4 -Find the factors of (x^2-16): (x+4)(x-4)/(x-4), x ≠ 4 -Cancel out like terms: x+4, x ≠ 4 -Make sure to include the domain restriction in your answer.
Multiple Choice Question 1: Identify the domain and range of a function
-Can be described in set builder notation, interval notation, or as an inequality. -Set Builder Notation Ex: -Domain: {x | x ∈ R} (means that all x values are real numbers) -Range: {y | y ∈ R} (means that all y values are real numbers) -{x | x ∈ Z} (means that all x values are integers). -{x | x ∈ N} (means that all x values are natural numbers). -{x | x ∈ Q} (means that all x values are rational numbers). -Interval Notation (bracket notation) Ex: -Domain: (−∞, ∞) or (0, ∞), meaning all real numbers. -Range: (−∞, ∞) or (−∞, 0), meaning all real numbers. -(−∞, 0) ∪ (0, ∞) (the ∪ means union of two sets) - ( ) are used if the numbers are excluded (and there is an open circle). - [ ] are used if the numbers are included (and there is a closed circle).
Multiple Choice Question 19: Identify key characteristics of a function
-Domain: inputs of a function (set of allowable x values). -Range: outputs of a function (set of allowable y values). -Intercepts: where the graph crosses the axes. -Increasing Interval: interval where gradient is positive. -Decreasing Interval: interval where gradient is negative. -Positive Internals: interval where y-values are positive. -Negative Internals: interval where y-values are negative. -Relative Extrema: turning points in a curve. -End Behavior: what the graph is doing x as approaches positive or negative infinity.
Multiple Choice Question 12: Identify end behaviors of a polynomial function
-End Behaviors: As x approaches either −∞ or ∞, f(x) approaches either −∞ or ∞ -Right side: looks at the leading coefficient: -If positive, right sides goes up -If negative, right side goes down -Left Side: looks at highest degree. -If even, left side matches right side and is a quadratic function. -If odd, left side is opposite the right side and is a cubic function.
Multiple Choice Question 13: Find the inverse of a function
-Inverse function: f^-1 is read, -1 is not an exponent. -f^-1(x) = INVERSE/of f(x). -(f(x))^-1 = RECIAPROCAL/of f(x) -> ex: 1/f(x) -Inverse relations: switch the x and y, (a,b) -> (b,a) -Ex: A=[{(1,5), (2,6), (3,7)}] -Inverse: B=[{(5,1), (6,2), (7,3)}] -The notation f^-1(x) only used if the inverse of the function is also a function. Otherwise, simply write the inverse in the form y=______________. -Ex of Inverse with Graphing: f(x)=2x+1 -> y=2x+1 -To find the inverse, first switch the x and the y: x=2y+1 -Then bring the x back to the other side: -2y=-x+1 (optional step: multiply it by -1 to get 2y=x-1). -Then divide -2 (or 2 if you did the optional step) to all 3 parts of the equation: -2y/-2=-1x/-2+1/-2 (optional step: 2y/2=1x/2-1/2). -You would get y=1/2x -1/2 -How do you know if the inverse will be a function? You could either use the horizontal line test (draw a horizontal line in the middle of the graph, if the line passes through two points or more, it is not a function) or figure it out algebraically (do the opposite compositions and if they reduce to just "x", the functions are inverses of each other). -Symbols for the algebraic way: f(x) and g(x) are inverses if and only if [f ∘ g](x)=x and [g ∘ f](x)=x -Ex: Let f(x)=3x+9 and g(x)=1/3x-3. Verify the compositions are functions of each other. -[f ∘ g](x) = f[g(x)] -Plug in g(x): f(1/3x-3) -Plug in f(x): 3(1/3x-3)+9 -Simplify: x-9+9 = x -Now check [g ∘ f](x): [g ∘ f](x) = g[f(x)] -Plug in f(x): g(3x+9) -Plug in g(x): 1/3(3x+9)-3 -Simplify: x+3-3 = x -Functions are inverses because [f ∘ g](x)=x = [g ∘ f](x)=x
Multiple Choice Question 7: Derive the equation of a parabola given the vertex, focus, and/or directrix
-Locus: set of all points that are equidistant from a given point (called focus) and a given line (called directrix). -The focus is always on the AoS (axis of symmetry) inside the curve of the parabola. -The directrix is always perpendicular to the AoS outside the curve of the parabola. -The vertex is always halfway between the focus and the directrix. -The vertical distance between the vertex and either focus or the directrix is always 1/4a -In a Function: -Form of Equation: y=a(x-h)^2 + k -Direction of opening: upward - a>0, downward - a<0 -Vertex: (h,k) -AoS: x = h -Focus: (h, k + 1/4a) -Directrix: y=k-1/4a -Vertex when given Focus and Directrix: (x-value of focus, directrix+y-value of focus/2). -In a NOT Function: -Form of Equation: x=a(y-k)^2 + h -Direction of opening: right - a>0, left - a<0 -Vertex: (h,k) -AoS: y = h -Focus: (h + 1/4a, k) -Directrix: x=h-1/4a -Vertex when given Focus and Directrix: (directrix+x-value of focus/2, y-value of focus). -Steps to find the equation of a parabola given the focus (inside of the parabola), directrix (outside of parabola) and vertex. -Sketch a picture. -Fill in what you know into vertex form of parabola. -Use other given information to find the "a" value. -Ex 1: Write vertex form of parabola described below: Vertex: (3,3) and Focus: (3,5). -Sketch a graph, and find the distance between the vertex and focus, which in this case is 2. So the equation to find a is 1/4a = 2. -Plug the values we know into vertex form. Since the graph is facing upward, it is a function: y=a(x-3)^2+3 -Find "a": 1/4a = 2 -> 1/4 = 2a -> 1/8 -Plug in the "a" value into the vertex form: y=1/8(x-3)^2+3. -Ex 2: Write vertex form of parabola described below: Vertex: (-2,4) and Directrix: x=-1. -Sketch a graph, and find the distance between the vertex and directrix, which in this case is 1. So the equation to find a is 1/4a = 1. -Plug the values we know into vertex form. Since the graph is facing to the left and the directrix is x=____, it is not a function: x=a(y-4)^2-2 -Find "a": 1/4a = 1 -> 1/4 = 1a -> 1/4 -Plug in the "a" value into the vertex form: x=1/4(y-4)^2-2. -Ex 3: Write vertex form of parabola described below: Focus: (1,2) and Directrix: x=-3. -We are going to use the formula to find the vertex when just given the focus and directrix. The focus is (1,2) and the directrix is x=-3 and it isn't a function, so we are going to add the directrix and x-value of the focus and divide that by 2, with the second coordinate being the y-value of the focus: [(-3+1)/2, 2)] -> (-1,2). -Sketch a graph, and find the distance between the vertex and either the directrix or focus, which in this case is 2. So the equation to find a is 1/4a = 2. -Plug the values we know into vertex form. Since the graph is facing to the left and the directrix is x=____, it is not a function: x=a(y-2)^2-1 -Find "a": 1/4a = 2 -> 1/4 = 2a -> 1/8 -Plug in the "a" value into the vertex form: x=1/8(y-2)^2-1.
Multiple Choice Question 6: Factor polynomial expressions. Part 1
-Once you factored completely, Set your factors equal to 0, then solve to find the solution. -Ex: (x+3)(2x-1)=0 - x+3=0 or 2x-1=0 - x=-3 or 2x=1 (divide 2 to both sides). - x=-3 or x=1/2 -Different types of factors: -Difference of Squares: a^2-b^2 = (a+b)(a-b) -Ex: 9x^2-4=0 -(3x+2)(3x-2) - x=-2/3 or x=2/3 -Factor by Grouping: -Ex: x^2-6x-27 -Find 2 numbers that multiply to get -27 and add to get -6: -9 and 3. -If there is a value in the quadratic is 1 (in this case it is because there is not other number in front of x^2), you can put the numbers (-9 and 3) in factor form to save time: (x-9) and (x+3). -Set the factors equal to 0 and solve: x-9=0 -> x=9, x+3=0 -> x=-3 -Answer: x=9 or x=-3 -Factor by Group when there is a value for "a": -Ex: 2x^2+5x-3 -Multiply 2 ("a" value) and -3 ("c" value) together (-6), and find two numbers that multiply to get -6 and add to get 5: 6 and -1. -Since there is a coefficient of x^2, you would plug in 6x and -1x into the quadratic as a replacement of 5x: 2x^2+6x-x-3=0. -Take values out to get the same numbers in the parentheses so that when you multiply them again, you get the same quadratic now (2x^2+6x-x-3): 2x(x+3)-1(x+3). -Now you have your factors: (2x-1) and (x+3). -Set them equal to 0 and get the solutions: x=1/2 and x=-3. -Factor by taking some numbers out: -Ex: 2x^2+8x=0 - 2x(x+4)=0 -Set the factors equal to 0 and solve: 2x=0, x+4=0 -Answer: x=0 or x=-4 -Remainder Theorem: -If we use synthetic division and get a remainder, the number is the value of the function at r, aka f(r). -Ex. Shown in picture Factor Theorem: -If we perform synthetic division and get a reminder of 0, then (x-r) is a factor of the polynomial. -if the remainder = 0, that's where it crosses the x-axis (x-intercept). -Ex: Show that (x+5) is a factor of x^3+2x^2-13x+10. Then find the remaining factors of the polynomial. - (x-r) -> x-(-5) - -5| 1 2 -13 10 + -5 15 -10 ------------- 1 -3 2 0 -> (x+5)(x^2-3x+2) -> (x+5)(x-2)(x-1) -Sum and Difference of 2 Cubes: -Sum of cubes: a^3 + b^3 = (a+b)(a^2-ab+b^2) -Difference of cubes: a^3 - b^3 = (a-b)(a^2+ab+b^2) -Use the acronym SOAP: S-Same O-Opposite A-Always P-Plus. -The first thing has the same sign as before, the second thing has the opposite sign, and the third thing is always plus. -Ex 1: x^3 + 27 - (x+3)(x^2-3x+9) -Ex 2: 16x^3 + 250y^3 - 2(8x^3 + 125y^3) - 2 (2x+5y)(4x^2-10xy+25y^2) -Letting U^2 take over: -If you get a polynomial that looks like this: ax^4+bx^2+x, you can plug in u^2 into x^4 and u into x^2. -Ex 1: 18x^4-21x^2+3, let u=x^2 - 18u^2-21u+3 - 3(6u^2-7u+1) -> 3(6u^2-6u-u+1) -> 3[6u(u-1)-1(u-1)] -> 3(6u-1)(u-1) -Plug x^2 back in: 3(6x^2-1)(x^2-1) -> 3(6x^2-1)(x+1)(x-1).
Multiple Choice Question 17: Evaluate a piece-wise function
-Piecewise-defined functions: written using two or more expressions, it's graph is often disjointed. -In the example shown, it has 3 functions to graph. -First, graph x for x ≤ -5. You do this by first graphing x, and you keep going down until you reach -5 (as an x value). Put a closed circle on that point (because the symbol is less than OR equal to) and draw a diagonal line. You do this because the x values in that function is supposed to be less than -5, so you first must go down to -5 to graph the function. -Second, graph -x for -5 < x ≤ 5. Because it is a -x, the line is going to go downwards. Graph the -5 < x first, by starting from zero and graphing x upwards (because the function is supposed to be going down). Once you reach -5 (as an x value), put an open circle because the symbol is just less than. Go back to 0 and graph x going down, stop at 5 (as an x value). Put a closed circle because the symbol is less than OR equal to). Draw a line between the open circle and the closed circle. This shows that for this function, the x values have to greater than -5, but less than 5. -Third, graph x for x > 5. Start at 0 and graph upwardly x (because the gradient is positive). Stop once you reach 5 (as the x value) and put an open circle on that point. Then draw a diagonal line upwards. This means that the x values must be greater than 5 in this function. -Some functions can be written as piece-wise defined or explicitly defined. -Ex of explicitly defined: y=|x|
IB Style Question 4: Analyze a polynomial function Multiple Choice Question 4: Divide polynomial expressions
-Simplifying Expressions Review: -Negative Exponents: a^-n = 1/a^n 1/a^-n = a^n -Product of Powers: a^m * a^n = a^m+n -Quotient of Powers: a^m/a^n = a^m-n -Properties of Powers: - (a^m)^n = a^m*n (ab)^m = (a^m)*(b^m) - (a/b)^n = (a^n)/(b^n), b ≠ 0 - (a/b)^-n = (b/a)^n or (b^m)/(a^n), a ≠ 0, a ≠ 0 -Ex 1: (3m^4 * n^-2)(-5mn)^2 -Distribute the ^2 to (-5mn): (25m^2 * n^2). -Multiply that product to (3m^4 * n^-2) by multiply 3 and 25 while combining the like terms for m^4 and m^2 and n^-2 and n^2: (3m^4 * n^-2)(25m^2 * n^2) = 75m^6. -Or you could have turned the ^-2 into (3m^4)/(n^2) * (25m^2 * n^2). -Ex 2: (-m^4)^3/(2m^2)^-2 -Simplify the ^-2: (-1m^4)^3 * (2m^2)^2 -> (-m^12)(4m^4) = -4m^16. -When adding polynomials, combine like terms. -When subtracting polynomials, distribute the negative and combine like terms. -When multiplying polynomials, use FOIL. -When dividing polynomials, use either simplifying, long division, or synthetic division. -To simplify, separate into fractions and then simplify: -Ex: Simplify (12p^3t^2r)-(21p^2qtr^2)-(9p^3tr)/(3p^2tr) -(12p^3t^2r)/(3p^2tr) - (21p^2qtr^2)/(3p^2tr) - (9p^3tr)/(3p^2tr) -For more complication division, use long division. -Ex: Shown in picture. -We can simplify the process more by using synthetic division, a procedure to divide a polynomial by a binomial using only the coefficients of the dividend and the value of "r" in the divisor (x-r) -In this process, the "x" in (x-r) must have a coefficient of 1 and must be "-r", it can't be "+r" [though you can manipulate it to make a (x-r) -> (x-(-r))]. -Ex: Shown in picture. -If a term is missing (like it goes from x^4, to x^2 and x^1), plug 0 into that missing term.
Multiple Choice Question 5: Identify the number and type of solutions to a quadratic
-The solutions of a graph the points where the graph crosses the x-axis. -Multiple names for those solutions: zeros, roots, and x-intercepts. -You can solve it by graphing the quadratic and identify the x-intercepts. -How to graph a quadratic: -Make sure the function is in ax^2+bx+c=0 -Ex: x^2+2x-8=0 -Find equation of axis of symmetry x=-b/2a -Ex: x=-(2)/2(1) = -2/2 = -1 -> x=-1, do not forget the "x=", otherwise you will get the problem wrong. -Find coordinates of the vertex by substituting the value of the AoS (axis of symmetry) into the equation (-b/2a,f(-b/2x) -> where x=-b/2a is the value of h and that value plugged into the quadratic is the value of k in (h,k). -Ex: f(x)=x^2+2x-8=0 -> y=(-1)^2+2(-1)-8 = -9 -> Vertex: (-1,-9). -Graph using the AoS, vertex, and the parent pattern. -The parent pattern is a pattern of points from the parent function [Ex in a quadratic function: (1,1), (2,4), (3,9)]. -To make life easier for you, if you need to find another point, use the x values of this pattern to find more points to graph. -Ex: Since the vertex is (-1,-9), go one to the right and notice the x value. In this case, going right from -1 would result in the x value being 0. -Next, plug 0 into the quadratic, which is x^2+2x-8=0. So that would be (0)^2+2(0)-8. Solve that and you will get your y-coordinate, which is -8. So the next point you can graph is (0,-8), and you can graph the opposite of that by going to the left from -1, which is -2. So another point you can graph is (-2, -8). -From there, you can go 2 from -1 to get another x-value (1), 4 from -1 to get another(3), and so on. And do the same process mentioned above to get your coordinates. -After you graph your quadratic, find where it crosses the x-axis. In this cash, it crosses at (0,-4) and (0,2). So your solutions are x=-4 or x=2. -If no points cross the x-axis, there are no real solutions. Do not just write "no solutions" because there are solutions, they just don't appear on the graph.
Multiple Choice Question 11: Evaluate a composition of functions
-[f ∘ g] (x) = f[g(x)] = composition f of g of x. -Ex 1: If f(x) = {(2,6), (9,4), (7,7), (0,-1)} and g(x) = {(7,0), (-1,7), (4,9), (8,2)}, find [f ∘ g](x). -Evaluate g(x) first: -g(7)=0 g(-1)=7 g(4)=9 g(8)=2 -Then use the range of g as the domain of f, and evaluate f(x): -f[g(7)] = f(0) or -1 -f[g(-1)] = f(7) or 7 -f[g(4)] = f(9) or 4 -f[g(8)] = f(2) or 6 -Answer: [f ∘ g](x) = {(7,-1), (-1,7), (4,4), (8,6)} -Ex 2: If f(x) = {(2,6), (9,4), (7,7), (0,-1)} and g(x) = {(7,0), (-1,7), (4,9), (8,2)}, find [g ∘ f](x). -Evaluate f(x) first: -f(2)=6 f(9)=4 f(7)=7 f(0)=-1 -Then use the range of f as the domain of g, and evaluate g(x): -g[f(2)] = f(6) or "no corresponding value" -g[f(9)] = f(4) or 9 -g[f(7)] = f(7) or 0 -g[f(0)] = f(-1) or 7 -Answer: [g ∘ f](x) = {(9,9), (7,0), (0,7)} -Ex 3: Find [f ∘ g](x) for f(x)=3x^2-x+4 and g(x)=2x-1 -[f ∘ g](x) = f[g(x)] -Replace g(x) with 2x-1: f(2x-1) -Substitute 2x-1 for x in f(x): 3(2x-1)^2-(2x-1)+4 -Solve using PEMDAS: 12x^2-14x+8
Multiple Choice Questions 9 and 15: Simplify complex expressions & Evaluate complex expressions IB Style Question 3: Simplify complex expressions
-i^2 = i*i = -1 -i: imaginary number, never a real number -imaginary number when squared gives a negative result. -the square root of -1 = √(−1) = i -By accepting i exists, we can solve things that needs the square root of a negative number. -Ex: What is the square root of -9? -√(9*-1) = √(9)*√(−1) = (3)√(−1) = 3i -3i^2 = (3i)(3i) = (3)^2 (i)^2 = 9*-1 = -9 -Imaginary numbers combined with real numbers make: complex numbers. -When adding complex numbers, combine like terms. -Ex: (3+4i) + (2+5i) = (5+9i) -When subtracting complex numbers, distribute the subtraction symbol to the second value and then combine like terms. -Ex: (2+5i) - (6-8i) - (2+5i)-6+8i = -4+13i -When multiplying complex numbers, use FOIL, then combine like terms. Remember that i*i = -1 - (4-3i)(-5-7i) - -20-28i+15i+21(-1) - -41-13i -When dividing complex numbers, multiply the fraction by i/i to get rid of the imaginary number in the denominator. -Ex: (2-3i)/4i * i/i = (2i-3(-1))/4(-1) -Solve, then rewrite the fraction and place the negative at the front with the opposite sign in the parentheses to the fractions: (2i+3)/-4 = -(3/4 + 2i/4) -Reduce 2i/4 to 1/2 i to get your answer: -3/4 + 1/2 i -If your dividing by a complex number with a real number portion, multiply the fraction by the denominator's conjugate. -Ex: (5)/(2+i) * (2-i)/(2-i) = (10-5i)/(4-(-1)) -Simplify: (10-5i)/(4+1) = (10-5i)/(5) = 2-i -Pattern of i -There is a pattern of the imaginary number i that repeats with 4 values in a cycle (1, i, -1, -i) - i^0=1 i^1=i i^2=-1 i^3=-i -This means you can evaluate any power of i by dividing the exponent by 4 and the remainder you get is equal to the power of i (which gives your answer). -Ex: i^37 -37/4 = 9 with a remainder of 1 - i^37 = i^1 = i -So, i^37 = i -If there was no remainder, it would be i^0, or 1
Overview of what to know
Part 1 - Multiple Choice: 1-Identify the domain and range of a function 2-Perform operations on functions 3-Simplify rational expressions 4-Divide polynomial expressions 5-Identify the number and type of solutions to a quadratic 6-Factor polynomial expressions 7-Derive the equation of a parabola given the vertex, focus, and/or directrix 8-Simplify radical expressions 9-Simplify complex expressions 10-Analyze a function 11-Evaluate a composition of functions 12-Identify end behaviors of a polynomial function 13-Find the inverse of a function 14-Rewrite an expression by completing the square 15-Evaluate complex expressions 16-Find the average rate of change over an interval of a function 17-Evaluate a piece-wise function 18-Identify transforming values of a quadratic equation given the graph 19-Identify key characteristics of a function 20-Identify an appropriate function for a real-world scenario 21-Identify an equation of a polynomial given a graph 22-Solve a radical equation Part 2 - IB Style Questions: 1-Analyze and rewrite the form of quadratic functions 2-Analyze functions 3-Simplify complex expressions 4-Analyze a polynomial function
