Notes on 2.5 Graphing Techniques: Transformations and 2.6 Mathematical Models
2.5 Graphing Techniques: Transformations
Type of function discussion (based on Page 2): If the x-values correspond to cube roots of the y-values, the relation is a cubic function. Specifically, if x = \sqrt[3]{y}, then y = x^3, so the graph is the cubic function y = x^3. An example in the notes uses the fact that the cube root of a value is the inverse of squaring, illustrating the idea that certain coordinate relationships imply specific standard forms (e.g., cube root vs. cube).
Desmos and graphing practice (Page 3): A Desmos example demonstrates transforming and combining expressions like \(x+3)^2\) and constant terms to obtain shifted or reflected parabolas. The notes hint at vertical shifts, horizontal shifts, and vertical reflections through explicit forms such as \(y = -((x+3)^2 + 5)\).
Transformation summary (Page 4): Core rules used to graph transformations of a base function f. While some phrasing in the transcript is a bit inconsistent, the standard, intentional forms to know are:
Vertical shifts: \ y = f(x) + k, \ k > 0; \ y = f(x) - k, \ k > 0
Vertical stretching/compression: \ y = a f(x), \ a > 0; stretching if a > 1, compression if 0 < a < 1
Horizontal shifts: \ y = f(x + h), \ h > 0 (shift left by h); \ y = f(x - h), \ h > 0 (shift right by h)
Horizontal scaling: \ y = f(a x), \ a > 0 (compression if a > 1; stretch if 0 < a < 1)
Reflections: \ Reflect about the x-axis: \ y = -f(x) ; \ Reflect about the y-axis: \ y = f(-x)
Functional changes (replacing x or scaling):
• Add or subtract a constant to the function: \ y = f(x) + k, \ y = f(x) - k
• Horizontal shift via replacing x: \ y = f(x + h), \ y = f(x - h)
• Horizontal scaling via replacing x with a scaled input: \ y = f(a x), \ a > 0
• Vertical scaling via multiplying the function: \ y = a f(x), \ a > 0
• Reflection via multiplying the output by -1: \ y = - f(x)
• Reflection via replacing x with -x: \ y = f(-x)
Quadratic transformations (Page 5): The standard form for a translated and stretched/shrunk quadratic is
[ f(x) = a (x - h)^2 + k ]a > 0 yields a vertical stretch; a < 0 yields a reflection about the x-axis and a vertical stretch/shrink depending on |a|.
h is a horizontal translation (to the right if h > 0, to the left if h < 0).
k is a vertical translation (up if k > 0, down if k < 0).
Examples:
Horizontal translation: ( f(x) = (x - h)^2 )
Vertical translation: ( f(x) = x^2 + k )
Reflection about the x-axis: when a < 0, e.g. ( f(x) = - (x - h)^2 + k )
Vertical stretch/compression: ( f(x) = a x^2 ) with a > 0 (stretch if a > 1, compression if 0 < a < 1)
Concrete transformation examples (Page 6): Common composed transforms include:
( y = 3(x^2 + 2) ) which equals ( y = 3 x^2 + 6 ): a vertical stretch by 3 plus a vertical shift up by 6.
Other listed forms demonstrate combining shifts with stretches, such as ( y = (x - 3)^4 ) or variations like ( y = (x+1) \,).
The take-away is that internal changes to x (inside the parentheses) produce horizontal shifts, while multiplying outside or altering the quadratic’s coefficient produce vertical scaling and shifts.
Sketching and writing equations (Pages 7–9):
Sketch tasks illustrate applying transformations to basic functions and then writing equations to represent the graphs after those transformations. A specific example in the notes shows f(x) = (-4)^3 + 5, which actually evaluates to a constant value (since (-4)^3 = -64), giving f(x) = -59 for all x; the graph would be a horizontal line at y = -59. This underscores the idea that some transformations can yield constant functions if the expression is constant.
Other items request writing equations for graphs that have undergone absolute-value and radical-style transformations (e.g., y = |x|, y = |x - a| + b, square-root forms like y = sqrt(1 - x) with shifts/reflections). The key idea is to recognize the basic form and apply horizontal/vertical shifts, reflections, and stretches/compressions accordingly.
Transformations of inverse and root functions (Page 12–13):
Example workflow for the square-root function:
1) Start with y = sqrt(x).
2) Apply a horizontal shift: y = sqrt(x + 1) (shift left by 1).
3) Apply a reflection about the y-axis if indicated: y = sqrt(1 - x) (reflect across the y-axis from the prior form).
4) Apply a vertical shift: y = sqrt(1 - x) + 2.General guideline on order (recommended in the notes):
1) Horizontal shifts, 2) Reflections, 3) Compressions/stretch, 4) Vertical shifts.“YOU TRY” exercises (Page 13) illustrate applying a sequence of simple transformations to reciprocal function forms, for example starting from y = 1/x and performing a rightward shift, then a vertical stretch, then a vertical shift to obtain an updated formula such as
[ y = \frac{3}{x-2} + 1. ]
Quick reference for tasks labeled in the notes:
To graph a transformed f, identify the type of transform (shift, stretch, reflection, etc.) and apply in a sensible order, often starting with horizontal changes, then reflections, then horizontal stretches/compressions, and finally vertical shifts.
When given a new equation, recognize it as a transformed base function (e.g., base: sqrt(x), reciprocal: 1/x, absolute value: |x|, etc.) and apply the corresponding changes to the input or the output.
2.6 Mathematical Models
The modeling workflow (Pages 14–14 and onward):
1) Draw a diagram or be given a diagram. Identify the quantity to be maximized or minimized and write an equation for that quantity.
2) Find the constraint equation that ties the variables together.
3) Use the constraint equation to substitute into the max/min equation so it is in terms of a single variable.
4) Graph the max/min equation with an appropriate viewing window.
5) Find the max/min point using a calculator.
6) State the answer to the problem with units.Problem Type 1: Building a box to maximize volume
Scenario: A topless box is built from a 20 in by 12 in piece of cardboard by cutting equal squares from each corner and folding up the sides.
Let x be the side length of the corner squares. Then the height h = x, the width w = 20 - 2x, and the length l = 12 - 2x.
Volume function:
[ V(x) = x (20 - 2x) (12 - 2x) = x [240 - 64x + 4x^2] = 240x - 64x^2 + 4x^3. ]Domain determined by nonnegative dimensions: 0 < x < 6 (since 12 - 2x > 0 and 20 - 2x > 0, the stricter bound is x < 6).
Graphing the cubic on an appropriate window reveals a maximum at approximately
[ x \approx 2.4274, \quad V_{max} \approx 262.68 \text{ cubic inches}. ]Practical takeaway: The largest-volume open-top box from a 20 by 12 card occurs when the corner squares are about 2.43 inches on a side.
Problem Type 2: Maximizing area of a rectangle inside a shape (HW 2.6 #4)
The rectangle has one corner on quadrant I on the graph of y = f(x) with f(x) = 36 - x^2, the opposite corner at the origin, and the other two corners on the positive axes.
Express the area A as a function of x: width = x, height = f(x) = 36 - x^2, so
[ A(x) = x \cdot (36 - x^2) = 36x - x^3. ]Domain: require x > 0 and f(x) > 0, i.e. 0 < x < 6.
The maximum occurs at the vertex of A(x). Differentiating or using the vertex formula for a cubic here (or by completing the square method) yields the maximizing x:
[ x = \sqrt{12} \approx 3.4641, ]
and the maximum area is
[ A_{max} = x f(x) = (\sqrt{12})(36 - 12) = (\sqrt{12}) \cdot 24 \approx 83.138. ]
Problem Type 3: Minimizing distance from a point to a curve
Setup: Let P = (x, y) be a point on a curve given by y = g(x). Let d be the distance from P to the origin:
[ d = \sqrt{ x^2 + y^2 } = \sqrt{ x^2 + [g(x)]^2 }. ]To minimize d, minimize d^2 = x^2 + [g(x)]^2 (avoid the square root).
The solution shows an example where the minimizing x-values are approximately
[ x \approx \pm 2.74, ]
with corresponding points on the curve around (±2.74, ∓2.74) (as indicated by the solution notes). The minimum distance is then approximately
[ d_{min} \approx \sqrt{(2.74)^2 + (2.74)^2} \approx 3.88. ]Key idea: minimize the squared distance, then compute the actual distance if needed.
Problem Type 4: Traveling problems (time minimization with different speeds)
Scenario: An island is 2 miles offshore from point P on a straight shoreline. A town is 12 miles downshore from P. A traveler can row at 3 mph and walk at 5 mph.
Let x denote the distance along the shore from P to the landing point of the boat. Then:
Row distance from island to landing point: ( \sqrt{2^2 + x^2} = \sqrt{4 + x^2} ).
Time rowing: ( \frac{\sqrt{4 + x^2}}{3} ).
Walking distance from landing point to town: ( 12 - x ) miles.
Time walking: ( \frac{12 - x}{5} ).
Total travel time as a function of x:
[ T(x) = \frac{\sqrt{4 + x^2}}{3} + \frac{12 - x}{5}. ]Domain: 0 ≤ x ≤ 12 (landing somewhere along the shore segment from P toward the town).
Graph T(x) and locate the minimum time using a calculator or optimization tool. The notes illustrate solving this optimization numerically and identifying the minimum time value and corresponding x.
Desmos/graphing solutions (recap)
The notes repeatedly show that many of these problems are solved by graphing the transformed functions in Desmos or a calculator, then identifying maximum/minimum points and reading off coordinates.
Summary of the modeling approach and connections to prior topics
Transformations provide the tools to model how graphs change under shifts, stretches, and reflections, which are essential when building mathematical models that depend on constraints and optimization.
The mathematical-model workflow ties directly to optimization problems often posed in real-world contexts (volume, area, distance, travel time). Recognizing the right objective and constraint is key to setting up solvable equations.
Notation recap (LaTeX-friendly):
Function transforms: ( y = f(x) + k ), ( y = f(x) - k ), ( y = a f(x) ), ( y = f(a x) ), ( y = f(-x) ), ( y = - f(x) ), etc.
Quadratic form with translations: ( f(x) = a (x - h)^2 + k ).
Area/volume: ( V(x) = \text{width} \times \text{length} \times \text{height} ), e.g. for the box problem, ( V(x) = x (20 - 2x) (12 - 2x) ).
Distance: ( d(x) = \sqrt{ x^2 + [f(x)]^2 } ) and ( d^2(x) = x^2 + [f(x)]^2 ).
Time: ( T(x) = \dfrac{\text{row distance}}{\text{row speed}} + \dfrac{\text{walk distance}}{\text{walk speed}}. )
Overall takeaway
The chapters on graphing techniques and mathematical models emphasize using graphing tools and algebraic substitutions to turn real-world constraints into tractable optimization problems. The core ideas are to identify the quantity to maximize/minimize, formulate a constraint-driven objective, reduce to a single-variable function, and use graphing/calculation to locate extrema with attention to units and domain. The provided examples (box volume, rectangle area, distance to a curve, and travel time) illustrate these steps in concrete terms.