FIND VERTEX OF ABSOLUTE VALUE EQUATIONS
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Additional Investigations
Distance to a Point in a Rectangle: Find an equation that relates segments inside a rectangle.
Intersecting Chords: Find an equation to relate chords of a circle.
Edge, Face, Vertex: Find an equation that relates the number of edges to the number of faces and the number of vertices in three-dimensional objects.
Area of a Kite: Find an equation that relates the area of a kite to its diagonals.
Segments on a Secant: Find an equation that relates segments on a secant.
Twice Reflected over Intersecting Lines: Find an equation which relates the angles formed by intersecting lines and points reflected over these lines.
Twice Reflected over Parallel Lines: Find an equation that relates the distance between points reflected over parallel lines.
Varied Slopes and Intercepts: Discover what happens to the graph when you vary the values for m and b in the equation y = mx + b.
Unusual Property: Can
?
Burglar Alarm: Determine how much money a company should charge for a burglar alarm system.
Composite Functions: Determine the composite function of two distinct functions.
Catching Up in Savings: How many years will it take Sally to have as much money as Judy?
Swapping Coordinates: What happens when you switch the x and y coordinates of a function?
Solutions in a System: What is the largest number of solutions possible in a given system of equations?
Input Equals Output: Find two functions so that the input of one is always equal to the output of the other.
In Between: Find a number such that its square is between two given numbers.
Crossing Curves: Find where a given linear function crosses a given quadratic function.
Fractional Forms of Repeating Decimals: Use a system of equations to determine the fractional form of a repeating decimal.
Zero Coefficients: In the equation Ax + By = C, what happens when A or B or C is zero?
Changing Intercepts: Discover what happens to the graph when you vary the values for n and b in the equation g(n) = n2 + b.
Triangle from a Line: The line Ax + By = C will form a triangle with the x and y axis for what values of A, B, and C?
Does 0=0?: What happens when two equations in the form y = mx + b are set equal to each other?
Finding the Maximum: Find a shortcut to find the x-coordinate of the maximum or minimum of a parabola.
Minimum Criteria: Given a set number of points on the coordinate plane, how many lines or parabolas can you make through these points?
Fundamental Theorem of Algebra: What is the largest number of times a function can cross the x-axis? How about the smallest number of times?
Symmetry of Polynomial Functions: Describe the symmetry of a function.
Using Roots to Generate an Equation: Find an equation of a parabola given the point(s) where it crosses the x-axis.
Exponential Functions: When is a function in the form y=abx increasing or decreasing?
Absolute Value Functions and Transformations: Describe how the graph of an absolute value function changes when you modify a, b, and c.
Reciprocal Functions: Describe any patterns you find when comparing a function to its reciprocal function.
Discriminant: How can the discriminant help you determine how many times the graph will cross the x-axis?
Forming a Square: Find four equations whose graphs are oblique (slanted) lines that will intersect to form the vertices of a square.
Sliding Graph: Examine the graphs of 2x + 3y = A.
Shifting Parabolas: Examine graphs of y=x2 + bx + 6, where b is any real number.
The Life of a Natural Number: Determine the behavior of natural numbers given that even and odd numbers act differently.
Earthquake Intensity: If a tremor can be thought of as a function of the form A(x - x2) in the interval(0, 1), examine quakes which are composed of a series of tremors.
Heron's Area: Explore the world of Heron triangles - triangles with sides of integral length and integral area.
Piano Frequency: Is the given relation comparing key position and frequency actually a function?
Altered State: Make a false equation correct by alternating the "punctuation" on the left side of the equation.
Comparing Temperatures: Use data to derive an equation that relates Celsius to Fahrenheit, Celsius to Kelvin, and Kelvin to Fahrenheit.
Pricing Pizzas: What type of function (linear, quadratic, or exponential) would you create to describe the cost of the pizza vs. the number of toppings?
Population Changes: Generate a function that will describe the population of your state vs. time.
Units of Speed: Find an equation that will compare speed units of miles per hour to kilometers per hour.
Keith's Pocket: How much money did Keith originally have in his pocket?
Symbolic Algebra: Determine which algebraic expression is greater.
Remainders and Factors: Determine the relationship of a group of positive integers that has 2 common divisors.
Reciprocal Sums: Given the sum of two positive integers, determine the smallest possible sum of their reciprocals.
Three Daughters: Determine the age of each daughter.
Piggy Savings: When will Parneshia's bank have twice as much money as John's?
The Frog Race: Two frogs have a race. Determine which frog wins the race.
Walking Rates: Determine the distance and time required for Amel to overtake Bonne, Amel's younger brother.
Counting Coins: How many coins does Kate have?
How Did We Do?: In a mathematics contest, how many of your team's answers were incorrect and how many problems were unanswered?
Formulaic Properties: Find values of x, y, and z, so that a given expression equals 18.
Legs and Seats: How many more legs than seats are in the leftover Seats 'R' Us factory inventory?
The Gonzalez Family: How old is each of the Gonzalez children?
The Game Show: Determine what each question is worth on a television game show.
Collecting Baseball Cards: On what day will Mike have exactly 100 baseball cards?
Lucky Find: How much money did Karen have before she found the $2?
Choosing Salaries: Your principal wants to hire you to work for her for ten days. Determine which way would you earn the most money.
Unknown Distance: Determine the distance between two points.
Integer Function: Find the smallest integer for which an expression results in an integer.
Nails in a Pail: Determine how much a pail and nails weigh.
Bouncing Ball: How many times does the special bouncing ball hit the ground?
Number Machine: What will the machine give me if I put in 100?
Summing to 100: Use the numbers from the given list no more than once to obtain a sum of 100.
Largest Combination: Find the largest possible value of a given expression.
Family Math: How old are a mother and son?
Double Trouble: Find a number that fits the description.
Judy's Gift: How much money did Judy give Dave?
Double Dollars: What is the fewest number of years until an investment doubles in value?
Sales Options: Which salary option should Mrs. Lee, the top salesperson in the company, take?
Library Fines: Help Minerva sort out her charges from the school library.
Going to the Movies: Predict the cost of the movie tickets.
Phone Tree: Determine the structure of a telephone tree used by eighth graders in Jefferson County who offered to help when the river flooded their town.
Paper Folding: Determine the thickness of folded paper.
Palindrome Odometer: Determine how fast a bus was traveling during two hours of time.
Mystery Number: What is the mystery number that fits the description?
Intersecting Chords: Find an equation to relate chords of a circle.
Edge, Face, Vertex: Find an equation that relates the number of edges to the number of faces and the number of vertices in three-dimensional objects.
Area of a Kite: Find an equation that relates the area of a kite to its diagonals.
Segments on a Secant: Find an equation that relates segments on a secant.
Twice Reflected over Intersecting Lines: Find an equation which relates the angles formed by intersecting lines and points reflected over these lines.
Twice Reflected over Parallel Lines: Find an equation that relates the distance between points reflected over parallel lines.
Varied Slopes and Intercepts: Discover what happens to the graph when you vary the values for m and b in the equation y = mx + b.
Unusual Property: Can
? Burglar Alarm: Determine how much money a company should charge for a burglar alarm system.
Composite Functions: Determine the composite function of two distinct functions.
Catching Up in Savings: How many years will it take Sally to have as much money as Judy?
Swapping Coordinates: What happens when you switch the x and y coordinates of a function?
Solutions in a System: What is the largest number of solutions possible in a given system of equations?
Input Equals Output: Find two functions so that the input of one is always equal to the output of the other.
In Between: Find a number such that its square is between two given numbers.
Crossing Curves: Find where a given linear function crosses a given quadratic function.
Fractional Forms of Repeating Decimals: Use a system of equations to determine the fractional form of a repeating decimal.
Zero Coefficients: In the equation Ax + By = C, what happens when A or B or C is zero?
Changing Intercepts: Discover what happens to the graph when you vary the values for n and b in the equation g(n) = n2 + b.
Triangle from a Line: The line Ax + By = C will form a triangle with the x and y axis for what values of A, B, and C?
Does 0=0?: What happens when two equations in the form y = mx + b are set equal to each other?
Finding the Maximum: Find a shortcut to find the x-coordinate of the maximum or minimum of a parabola.
Minimum Criteria: Given a set number of points on the coordinate plane, how many lines or parabolas can you make through these points?
Fundamental Theorem of Algebra: What is the largest number of times a function can cross the x-axis? How about the smallest number of times?
Symmetry of Polynomial Functions: Describe the symmetry of a function.
Using Roots to Generate an Equation: Find an equation of a parabola given the point(s) where it crosses the x-axis.
Exponential Functions: When is a function in the form y=abx increasing or decreasing?
Absolute Value Functions and Transformations: Describe how the graph of an absolute value function changes when you modify a, b, and c.
Reciprocal Functions: Describe any patterns you find when comparing a function to its reciprocal function.
Discriminant: How can the discriminant help you determine how many times the graph will cross the x-axis?
Forming a Square: Find four equations whose graphs are oblique (slanted) lines that will intersect to form the vertices of a square.
Sliding Graph: Examine the graphs of 2x + 3y = A.
Shifting Parabolas: Examine graphs of y=x2 + bx + 6, where b is any real number.
The Life of a Natural Number: Determine the behavior of natural numbers given that even and odd numbers act differently.
Earthquake Intensity: If a tremor can be thought of as a function of the form A(x - x2) in the interval(0, 1), examine quakes which are composed of a series of tremors.
Heron's Area: Explore the world of Heron triangles - triangles with sides of integral length and integral area.
Piano Frequency: Is the given relation comparing key position and frequency actually a function?
Altered State: Make a false equation correct by alternating the "punctuation" on the left side of the equation.
Comparing Temperatures: Use data to derive an equation that relates Celsius to Fahrenheit, Celsius to Kelvin, and Kelvin to Fahrenheit.
Pricing Pizzas: What type of function (linear, quadratic, or exponential) would you create to describe the cost of the pizza vs. the number of toppings?
Population Changes: Generate a function that will describe the population of your state vs. time.
Units of Speed: Find an equation that will compare speed units of miles per hour to kilometers per hour.
Keith's Pocket: How much money did Keith originally have in his pocket?
Symbolic Algebra: Determine which algebraic expression is greater.
Remainders and Factors: Determine the relationship of a group of positive integers that has 2 common divisors.
Reciprocal Sums: Given the sum of two positive integers, determine the smallest possible sum of their reciprocals.
Three Daughters: Determine the age of each daughter.
Piggy Savings: When will Parneshia's bank have twice as much money as John's?
The Frog Race: Two frogs have a race. Determine which frog wins the race.
Walking Rates: Determine the distance and time required for Amel to overtake Bonne, Amel's younger brother.
Counting Coins: How many coins does Kate have?
How Did We Do?: In a mathematics contest, how many of your team's answers were incorrect and how many problems were unanswered?
Formulaic Properties: Find values of x, y, and z, so that a given expression equals 18.
Legs and Seats: How many more legs than seats are in the leftover Seats 'R' Us factory inventory?
The Gonzalez Family: How old is each of the Gonzalez children?
The Game Show: Determine what each question is worth on a television game show.
Collecting Baseball Cards: On what day will Mike have exactly 100 baseball cards?
Lucky Find: How much money did Karen have before she found the $2?
Choosing Salaries: Your principal wants to hire you to work for her for ten days. Determine which way would you earn the most money.
Unknown Distance: Determine the distance between two points.
Integer Function: Find the smallest integer for which an expression results in an integer.
Nails in a Pail: Determine how much a pail and nails weigh.
Bouncing Ball: How many times does the special bouncing ball hit the ground?
Number Machine: What will the machine give me if I put in 100?
Summing to 100: Use the numbers from the given list no more than once to obtain a sum of 100.
Largest Combination: Find the largest possible value of a given expression.
Family Math: How old are a mother and son?
Double Trouble: Find a number that fits the description.
Judy's Gift: How much money did Judy give Dave?
Double Dollars: What is the fewest number of years until an investment doubles in value?
Sales Options: Which salary option should Mrs. Lee, the top salesperson in the company, take?
Library Fines: Help Minerva sort out her charges from the school library.
Going to the Movies: Predict the cost of the movie tickets.
Phone Tree: Determine the structure of a telephone tree used by eighth graders in Jefferson County who offered to help when the river flooded their town.
Paper Folding: Determine the thickness of folded paper.
Palindrome Odometer: Determine how fast a bus was traveling during two hours of time.
Mystery Number: What is the mystery number that fits the description?
Taking Classes: Given certain parameters, how many possible students can take math and English, science and English, and math and science?
Bouncing Ball: Use a Computer BasedLaboratory (CBL) device to investigate the motion of a bouncing ball.
Chill Out: Use a Computer BasedLaboratory (CBL) device to investigate what happens to the temperature of a cupof hot water while time is passing.
It's Getting Darker: Use a ComputerBased Laboratory (CBL) device to investigatehow the intensity of light changes with the distance from a light source such asa bulb or a flashlight.
Don't Push It: Use a Computer BasedLaboratory (CBL) device toinvestigate how the gas pressure changes with the volume.
Cereal Bars: Solve a system of equations to determine the units of ingredients needed to make a cereal bar.
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