Algebra Parent Functions Flashcards

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Questions and Answers

What is the linear function parent?

  • f(x)=x (correct)
  • f(x)=2^x
  • f(x)=x^2
  • f(x)=√x

What is the linear graphing form?

y-y1=m(x-x1)

What is the quadratic parent function?

  • f(x)=∛x
  • f(x)=2^x
  • f(x)=x^2 (correct)
  • f(x)=x^3

What is the quadratic graphing form?

<p>y=a(x-h)^2+k</p>
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What is the absolute value function parent?

<p>f(x)= |x| (D)</p>
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What is the absolute value graphing form?

<p>y=a|x-h|+k</p>
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What is the square root function parent?

<p>f(x)=√x (A)</p>
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What is the square root graphing form?

<p>y=a√(x-h)+k</p>
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What is the cubic function parent?

<p>f(x)=x^3 (B)</p>
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What is the cubic function graphing form?

<p>y=a(x-h)^3+k</p>
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What is the cube root function parent?

<p>f(x)=∛x (B)</p>
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What is the cube root graphing form?

<p>y=a∛(x-h)+k</p>
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What is the exponential function parent?

<p>f(x)=2^x (A)</p>
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What is the reciprocal parent function?

<p>f(x)= 1/x (B)</p>
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What is a function?

<p>A relation between a set of inputs and outputs where each input has exactly one output.</p>
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What does 'a' tell you in an equation?

<p>The vertical stretch or compression factor and if the graph has been reflected over the x-axis.</p>
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What does 'h' tell you in an equation?

<p>The horizontal shift right or left.</p>
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What does 'k' tell you in an equation?

<p>The vertical shift up or down.</p>
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What are transformations in graphing?

<p>Changes made to a graph that doesn't alter the type of graph (A)</p>
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What is absolute value?

<p>Distance of a point from zero.</p>
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What does 'range' mean in mathematics?

<p>All possible y values for a function.</p>
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What does 'domain' mean in mathematics?

<p>All possible x values for a function.</p>
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Flashcards

Linear Function Parent

f(x) = x, a straight line through the origin.

Quadratic Function Parent

f(x) = x², a parabola opening upwards.

Absolute Value Function Parent

f(x) = |x|, a "V" shaped graph showing distance from zero.

Square Root Function Parent

f(x) = √x, a curve starting at the origin, increasing gradually.

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Cubic Function Parent

f(x) = x³, an S-shaped curve crossing through the origin.

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Cube Root Function Parent

f(x) = ∛x, a curve that exists for all real numbers.

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Exponential Function Parent

f(x) = 2^x shows rapid growth for positive x.

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Reciprocal Function Parent

f(x) = 1/x, two hyperbolas in opposite quadrants.

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Linear Graphing Form

y - y₁ = m(x - x₁), defines a straight line.

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Quadratic Graphing Form

y = a(x - h)² + k, transformations via vertex (h, k).

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Absolute Value Graphing Form

y = a|x - h| + k, absolute value function transformations.

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Square Root Graphing Form

y = a√(x - h) + k, square root transformations.

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Cubic Graphing Form

y = a(x - h)³ + k, cubic function transformations.

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Cube Root Graphing Form

y = a∛(x - h) + k, cube root transformations.

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Function Definition

Links inputs to unique outputs, one output per input.

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One-to-One Function

Each input has a unique output, and vice versa.

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Domain

All possible x values that give defined outputs.

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Range

All possible y values a function can produce.

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Transformations

Graph changes like stretches, compressions, reflections.

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Absolute Value

Distance from zero.

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Parameter 'a'

Vertical stretch/shrink; reflection over x-axis if negative.

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Parameter 'h'

Horizontal shift (right if positive, left if negative).

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Parameter 'k'

Vertical shift (up if positive, down if negative).

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Study Notes

Parent Functions Overview

  • Linear Function Parent: Defined by the equation f(x) = x, representing a straight line through the origin.
  • Quadratic Function Parent: Characterized by f(x) = x², resulting in a parabolic curve that opens upwards.
  • Absolute Value Function Parent: Expressed as f(x) = |x|, displaying a "V" shape graph, indicating distance from zero.
  • Square Root Function Parent: Given by f(x) = √x, featuring a graph that starts at the origin and increases gradually.
  • Cubic Function Parent: Defined by f(x) = x³, producing an S-shaped curve crossing through the origin.
  • Cube Root Function Parent: Given by f(x) = ∛x, which produces a curve allowing all real numbers.
  • Exponential Function Parent: Represented by f(x) = 2^x, revealing rapid growth for positive x values.
  • Reciprocal Function Parent: Defined by f(x) = 1/x, featuring two hyperbolas in opposite quadrants.

Graphing Forms

  • Linear Graphing Form: y - y₁ = m(x - x₁), representing a line on a graph using a point and a slope.
  • Quadratic Graphing Form: y = a(x - h)² + k, providing transformations through vertex form with 'a', 'h', and 'k'.
  • Absolute Value Graphing Form: y = a|x - h| + k, indicating transformations of the absolute value function.
  • Square Root Graphing Form: y = a√(x - h) + k, involving vertical/horizontal stretches and shifts of the square root function.
  • Cubic Graphing Form: y = a(x - h)³ + k, translating cubic functions while adjusting its steepness.
  • Cube Root Graphing Form: y = a∛(x - h) + k, similar transformations applicable as in other parent functions.

Function Characteristics

  • Function Definition: A relation connecting inputs to unique outputs, ensuring each input corresponds to only one output.
  • One-to-One Function: Each input maps to a unique output, and vice versa, ensuring no value is repeated in either domain or range.
  • Domain: Represents all possible x values that can be plugged into a function without resulting in undefined outputs.
  • Range: Encompasses all potential y values produced by the function as outputs across the domain.

Transformations and Features

  • Transformations: Graph alterations such as stretching, compressing, or reflecting, that retain the type of graph.
  • Absolute Value: Defines the numerical distance of any value from zero on the number line.
  • Parameter 'a': Indicates vertical stretch/shrink of the graph and shows reflection across the x-axis if negative.
  • Parameter 'h': Represents horizontal shifts, determining how far right (positive) or left (negative) the graph moves.
  • Parameter 'k': Determines vertical shifts, indicating movement up (positive) or down (negative) from the original graph position.

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