Transformations of Graphs and Quadratic Functions in Business Calculus

Transformations of Graphs

Vertical and Horizontal Translations

Transformations allow us to shift the graph of a function vertically or horizontally. These shifts are determined by adding or subtracting constants to the function or its variable.

To graph:	Shift f(x) by c units:
f(x) + c	upward
f(x) - c	downward
f(x + c)	to the left
f(x - c)	to the right

• Vertical translation: Adding/subtracting a constant outside the function shifts the graph up or down.

• Horizontal translation: Adding/subtracting a constant inside the function argument shifts the graph left or right.

Example: To graph $y = (x-1)^3 - 3$, start with $y = x^3$, shift right by 1 unit, and down by 3 units.

Reflections

Reflections flip the graph of a function across an axis.

• Reflection across the x-axis: $y = -f(x)$ is the reflection of $y = f(x)$ across the x-axis.

• Reflection across the y-axis: $y = f(-x)$ is the reflection of $y = f(x)$ across the y-axis.

Example: Starting with $y = \sqrt{x}$, graph $y = -\sqrt{x} + 1$, $y = -\sqrt{x-1}$, and $y = \sqrt{1-x}$ by applying reflections and translations.

Vertical Stretching and Compression

Multiplying a function by a constant $a$ affects its vertical stretch or compression.

• Vertical stretch: If $|a| > 1$, the graph is stretched vertically.

• Vertical compression: If $0 < |a| < 1$, the graph is compressed vertically.

Example: For $y = 2x^3$ and $y = \frac{1}{2}x^3$, the graph of $y = x^3$ is stretched and compressed, respectively.

Quadratic Functions

Definition and Standard Form

A quadratic function is a polynomial of degree 2, generally written as:

• $f(x) = ax^2 + bx + c$, where $a, b, c$ are real numbers and $a \neq 0$.

Quadratic functions graph as parabolas.

Vertex Form and Properties

The standard quadratic function can be rewritten in vertex form:

• $f(x) = a(x-h)^2 + k$

• where $h = -\frac{b}{2a}$ and $k = f(h) = \frac{4ac - b^2}{4a}$

Properties of the parabola:

• Vertex: $(h, k)$

• Axis of symmetry: $x = h$

• Direction: Opens upward if $a > 0$ (minimum at vertex), downward if $a < 0$ (maximum at vertex)

• y-intercept: $y = c$

• x-intercepts: Solve $f(x) = 0$

• Vertical stretch/compression: As with other functions, $|a| > 1$ stretches, $0 < |a| < 1$ compresses

Example: For $f(x) = 2x^2 + 12x + 10$:

• Domain: $(-\infty, +\infty)$

• Vertex: $h = -\frac{12}{4} = -3$, $k = f(-3) = -8$

• Axis: $x = -3$

• y-intercept: $y = 10$

• x-intercepts: Solve $2x^2 + 12x + 10 = 0$

• Range: $[-8, \infty)$

Graphing Parabolas with Given Vertex and Point

To find a quadratic function with a specified vertex $(h, k)$ and passing through a point $(x_1, y_1)$:

• Use $f(x) = a(x-h)^2 + k$

• Plug in $(x_1, y_1)$ to solve for $a$

Example: Vertex $(2, -3)$, point $(1, 0)$:

• $0 = a(1-2)^2 - 3 \implies a = 3$

• Function: $f(x) = 3(x-2)^2 - 3$

Applications: Cost, Revenue, and Profit Functions

Business Application of Quadratic Functions

Quadratic functions are used to model cost, revenue, and profit in business calculus.

• Cost function: $C(x)$, e.g., $C(x) = 12x + 32$

• Revenue function: $R(x)$, e.g., $R(x) = -4x^2 + 36x$

• Profit function: $P(x) = R(x) - C(x)$

Key concepts:

• Break-even quantity: The value of $x$ where $P(x) = 0$

• Maximum profit: The vertex of the profit function $P(x)$

Example: For $C(x) = 12x + 32$, $R(x) = -4x^2 + 36x$:

• Profit: $P(x) = -4x^2 + 24x - 32$

• Break-even: Solve $P(x) = 0 \implies x = 2$ and $x = 4$

• Maximum profit: Vertex at $x = 3$, $P(3) = 4$ ($4,000)

Quadratic Regression and Data Modeling

Fitting Quadratic Models to Data

Quadratic regression is used to fit a quadratic function to real-world data, such as age at first marriage over time.

Year (1900+t)	Age
1950	20.3
1960	20.3
1970	20.8
1980	22.0
1990	23.9
2000	25.1
2010	26.1
2018	27.8

• Plot the data using $t = 50$ for 1950, etc.

• Find the quadratic function $P(t)$ of best fit.

• Use $P(t)$ to approximate age in 2025 ($t = 125$).

Numerical solution: $P(t) = 0.0011t^2 - 0.0712t + 20.7555$

Vertex form modeling: Given vertex $(50, 20.3)$ and point $(118, 27.8)$, solve for $a$ in $f(t) = a(t-50)^2 + 20.3$ using the point.

Example: $f(118) = a(118-50)^2 + 20.3 = 27.8 \implies a \approx 0.01622$

Approximate for 2025: $f(125) \approx 29.0$ years

Additional info: Quadratic regression is a common technique in business calculus for modeling trends and making predictions based on historical data.