MODELING A CONTEXT FROM A GRAPH

43210 In addition to level 3.0 and above and beyond what was taught in class, the student may: - Make connection with other concepts in math. - Make connection with other content areas. Students will be able to construct, compare, and analyze function models and interpret and solve contextual problems. Function models: - absolute value - square root - cube root - piecewise Analyze multiple representations of functions using: - Key features - Translations - Parameters/limits of domain Students will be able to construct and compare function models and solve contextual problems. Function models: - linear - exponential - quadratic Illustrate the graphical effects of translations on function models using technology. With help from the teacher, the student has partial success with the unit content. Even with help, the student has no success with the unit content. Focus 11 - Learning Goal: Students will be able to construct, compare and analyze function models and interpret and solve contextual problems.

Find your function!  Each student needs a card with a either a graph or an equation.  Look at your card and think about what your card will look like as an equation or as a graph.  Hold up your card and mingle around the room to find your match.  When you find your match, stand next to that person.  (You may have to estimate some values on the graphs.)

Find your function!  What was your strategy when trying to find your graph or function match?  If you had come up with a function based only on looking at a graph (rather than having a function to match it to) what might some possible steps be?  Identify what type of function it is (linear, quadratic, cubic, exponential…)  Compare the graph to the parent function; look for transformations.  Check the equation by testing a couple of points that can be read from the graph.  Make a table of values of known ordered pairs.  (Turn in cards and go back to desks.)

Example 1  The relationship between the length of one of the legs, in feet, of an animal and its walking speed, in feet per second, can be modeled by the graph.  Note: This function ONLY applies to the walking speed not running speed. Obviously, a cheetah has shorter legs than a giraffe but can run much faster. However, in a walking race, the giraffe has the advantage.  A T-Rex’s leg length was 20 feet. What was the T-Rex’s speed in feet/second?

Example 1  What are the units involved in this problem?  Leg length = feet  Speed = feet/second  What type of function does this graph represent?  Square root function

Example 1

g(x) works.

Example 1 h(x) doesn’t work.

Example 1

Example 2  The cross section view of a deep river gorge is modeled by the graph where both height and distance are measured in miles.  How long is the bridge that spans the gorge from the point labeled (1, 0) to the other side?  How high above the bottom of the gorge is the bridge?

Example 2  What type of function can be represented by a graph like this?  What is a general form for this function type?  f(x) = ax 2 + bx + c  g(x) = a(x – h) 2 + k  h(x) = a(x – m)(x – n)  Which one should we use?  Use the 3 ordered pairs, substitute, and see which one works.

Example 2  Start with (0, 5) into f(x).  f(0) = a(0) 2 + b(0) + 5  f(0) = 5  This means that c = 5  Substitute (1, 0) and solve.  0 = a(1) 2 + b(1) + 5  a + b = -5

Example 2  Substitute (3, 2) and solve.  2 = a(3) 2 + b(3) + 5  -3 = 9a + 3b  We now have a system of equations to solve:  a + b = -5  9a + 3b = -3  (multiply the top equation by -3 so you can use linear combinations to solve)  -3a - 3b = 15  9a + 3b = -3  6a = 12  a = 2  2 + b = -5  b = -7  The equation is f(x) = 2x 2 -7x + 5.

Example 2  With the equation, f(x) = 2x 2 -7x + 5, we are now able to answer the question.  How long is the bridge that spans the gorge from the point labeled (1, 0) to the other side?  We need to find the other x-intercept.  Factor to solve.  (You already know that one part of the factored equation is x – 1.)

Example 2  f(x) = 2x 2 -7x + 5, in factored form is f(x) = (x – 1)(2x – 5).  The 2 nd x-intercept is at 2.5.  The bridge is the distance from 1 to 2.5.  The bridge is 1.5 miles long.

Example 2  With the equation, f(x) = 2x 2 -7x + 5, we are now able to answer the 2 nd question.  How high above the bottom of the gorge is the bridge?  This is the distance between the vertex and the x-axis.  Find the vertex.

Example 2  f(x) = 2x 2 -7x + 5  The x part of the vertex is found with x = -b / 2a.  x = 7 / 2(2)  x = 7 / 4  y = 2( 7 / 4 ) 2 – 7( 7 / 4 ) + 5  y = 1 1 / 8 Vertex ( 7 / 4, 1 1 / 8 )  The bottom of the gulch is 1 1 / 8 miles from the bridge.