Comprehensive Study Notes: Quadratic and Power Functions, Domain & Range, Function Transformations

Quadratic Functions

Standard Form of Quadratic Functions

Quadratic functions are fundamental in algebra and are typically written in the standard form: $f(x) = ax^2 + bx + c$, where a, b, and c are constants and $a \neq 0$. The graph of a quadratic function is a parabola.

• Quadratic term: $ax^2$; the coefficient a determines the direction and width of the parabola.

• Linear term: $bx$; affects the slope and position.

• Constant term: $c$; the y-intercept of the graph.

• If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

Example: $h(t) = -16t^2 + 80t + 96$ models the height of a thrown baseball, where $-16$ indicates the parabola opens downward.

Factored and Vertex Forms

Quadratic functions can also be written in other forms to reveal different properties:

• Factored form: $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the zeros (x-intercepts).

• Vertex form: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Example: $h(t) = -16(t - 6)(t + 1)$ shows zeros at $t = 6$ and $t = -1$. $h(t) = -16(t - 2.5)^2 + 196$ has vertex at $(2.5, 196)$.

Graphical Interpretation

Key features of quadratic graphs include zeros, vertex, and y-intercept. The vertex represents the maximum or minimum value depending on the sign of a.

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• The discriminant $D = b^2 - 4ac$ determines the nature of the solutions:

  • If $D > 0$, two real solutions.

  • If $D = 0$, one real (double) solution.

  • If $D < 0$, no real solutions (complex roots).

Power Functions

Positive Exponents

A power function is any function of the form $f(x) = kx^n$, where $k \neq 0$ and $n$ is a real number. The graph's shape depends on the exponent:

• Quadratic ($n=2$): U-shaped parabola.

• Cubic ($n=3$): Chair-shaped curve.

Example: The depth of a falling ball: $d(t) = 4.9t^2$.

Applications of Power Functions

Power functions are used in modeling real-world phenomena, such as stopping distances for vehicles.

Speed	Perception/Reaction Distance	Braking Distance	Overall Stopping Distance	Approx. Car Lengths (@15 ft)
30 mph	44 ft	45 ft	89 ft	6
40 mph	59 ft	80 ft	139 ft	9
50 mph	73 ft	125 ft	198 ft	14
60 mph	88 ft	180 ft	268 ft	18
70 mph	103 ft	245 ft	348 ft	23
80 mph	117 ft	320 ft	439 ft	29

Negative and Fractional Exponents

Power functions with negative exponents, such as $f(x) = x^{-1}$, have vertical and horizontal asymptotes. Fractional exponents, such as $f(x) = x^{1/2}$, represent root functions.

• For $n < 0$, $f(x)$ approaches zero as $x \to \infty$ and is undefined at $x = 0$.

• For $0 < n < 1$, $f(x)$ increases slowly as $x$ increases.

Domain and Range

Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Restrictions may occur due to division by zero or taking the square root of a negative number.

• For rational functions, exclude values that make the denominator zero.

• For root functions of even order, restrict to values where the radicand is non-negative.

Range of Functions

The range is the set of all possible output values (y-values). The range can be determined algebraically or by analyzing the graph.

• For quadratic functions, the range depends on the vertex and direction of opening.

• For root functions, the range is typically $[0, \infty)$ for even roots, unless shifted.

Function Transformations

Vertical and Horizontal Shifts

Transformations change the position and shape of a function's graph. Vertical shifts move the graph up or down, while horizontal shifts move it left or right.

• Vertical shift: $y = f(x) + c$ shifts up by $c$ units; $y = f(x) - c$ shifts down by $c$ units.

• Horizontal shift: $y = f(x - c)$ shifts right by $c$ units; $y = f(x + c)$ shifts left by $c$ units.

Vertical Stretch, Compression, and Reflection

Multiplying the function by a constant stretches or compresses it vertically. Reflecting the function over the x-axis changes the sign of all y-values.

• Vertical stretch: $y = af(x)$, $a > 1$ stretches; $0 < a < 1$ compresses.

• Reflection: $y = -f(x)$ reflects over the x-axis.

Composition and Inverse Functions

Composition of Functions

Composition involves applying one function to the result of another: $f(g(x))$. The inside function is $g(x)$, and the outside function is $f(x)$.

• Example: Converting Celsius to Fahrenheit: $F = 1.8C + 32$ can be viewed as composing two functions.

Inverse Functions

An inverse function "undoes" the action of the original function. If $f(x)$ and $g(x)$ are inverses, then $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the domain.

• To find the inverse, swap $x$ and $y$ in the equation and solve for $y$.

Factoring Quadratic Expressions

Factoring Methods

Quadratic expressions can be factored using sum-product, grouping, or special formulas:

• Sum-product: For $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.

• Perfect square: $x^2 + 2ax + a^2 = (x + a)^2$

• Difference of squares: $x^2 - a^2 = (x - a)(x + a)$

Example: Factor $x^2 - 81 = (x - 9)(x + 9)$

Summary Table: Quadratic Forms

Form	Equation	Key Features
Standard	$ax^2 + bx + c$	Vertex, y-intercept, direction
Factored	$a(x - r_1)(x - r_2)$	Zeros (x-intercepts)
Vertex	$a(x - h)^2 + k$	Vertex $(h, k)$, max/min value

Additional info: Academic context was added to clarify the relationship between forms, transformations, and applications of quadratic and power functions, as well as domain and range concepts.