Parent Functions Review Slides
Algebra 1 Review Core Function Families
f(x) = ?
Parent Functions
Visual Notes & Key Features
Mastering the four foundational Algebra 1 functions: Linear, Quadratic, Absolute Value, and Square Root.
Linear
\( f(x) = x \)
Quadratic
\( f(x) = x^2 \)
Absolute Value
\( f(x) = |x| \)
Square Root
\( f(x) = \sqrt{x} \)
What Is a Parent Function?
Algebra 1 Basics
Core Definition
The simplest, most basic equation in a family of functions before any changes or movements happen.
Think of it as the original starting shape. Later in Algebra 1, you learn to shift, slide, flip, or stretch it across the coordinate plane!
Domain (Inputs / \(x\))
All the possible \(x\)-values you can plug into the function that make mathematical sense.
Range (Outputs / \(y\))
All the resulting \(y\)-values that come out of the function when you graph it.
Anchor Point (Landmark)
The key reference point on the graph—often the vertex, starting endpoint, or origin \((0,0)\).
Key Goal: Recognize each equation by sight, picture its shape, and state its domain and range!
Family 1
Linear Function: \( f(x) = x \)
Straight-Line Graph
Slope \(m = 1\), \(y\)-intercept \((0,0)\)
Domain
All Real Numbers
\( -\infty < x < \infty \)
Range
All Real Numbers
\( -\infty < y < \infty \)
Table of Key Coordinates
\(x\)-2
\(x\)-1
\(x\)0
\(x\)1
\(x\)2
\(y\)-2
\(y\)-1
\(y\)0
\(y\)1
\(y\)2
Rate of Change: Constant slope of \(1\). For every \(1\) unit to the right, it rises exactly \(1\) unit up (\(\text{rise}/\text{run} = 1/1\)).
Algebra 1 Focus: The output always equals the input exactly.
Family 2
Quadratic Function: \( f(x) = x^2 \)
U-Shaped Parabola
Vertex at \((0,0)\), Opens Upward
Domain
All Real Numbers
You can square any number
Range
\( y \ge 0 \)
Outputs are zero or positive
Key Algebra 1 Features
- Vertex / Minimum: Lowest point on the graph is \((0,0)\).
- Line of Symmetry: The vertical line \(x = 0\) (the \(y\)-axis) cuts the graph into mirror halves.
- Why \(y \ge 0\)? Multiplying any real number by itself produces a positive number or zero.
(-2, 4)
(-1, 1)
(0, 0)
(1, 1)
(2, 4)
Algebra 1 Tip: Note how \((-2)^2 = 4\) and \(2^2 = 4\) both produce positive 4.
Family 3
Absolute Value: \( f(x) = |x| \)
Sharp V-Shaped Graph
Sharp Corner at \((0,0)\)
Domain
All Real Numbers
You can take \(|x|\) of any number
Range
\( y \ge 0 \)
Distance cannot be negative
Distance From Zero Concept
Absolute value represents distance on a number line from \(0\). Because distance is never negative, the entire graph stays at or above the \(x\)-axis.
Left Side: slope \(m = -1\)
Right Side: slope \(m = +1\)
Parabola vs. V-Shape: Parabola has a rounded curve; Absolute value has straight straight lines with a point.
Algebra 1 Tip: \((-3, 3)\) and \((3, 3)\) share the exact same height because \(|-3| = |3| = 3\).
Family 4
Square Root: \( f(x) = \sqrt{x} \)
Radical Curve / Starting Endpoint
Definite Starting Point at \((0,0)\)
Domain
\( x \ge 0 \)
No negative inputs under \(\sqrt{\phantom{x}}\)
Range
\( y \ge 0 \)
The principal square root is \(\ge 0\)
Use Perfect Squares to Graph!
(0, 0) Start
(1, 1) \(\sqrt{1} = 1\)
(4, 2) \(\sqrt{4} = 2\)
(9, 3) \(\sqrt{9} = 3\)
Why the domain restriction? In real numbers, you cannot take the square root of a negative number (e.g., \(\sqrt{-9}\) is not a real number because no real number times itself gives \(-9\)).
Algebra 1 Tip: There is NO graph to the left of the \(y\)-axis (\(x < 0\))!
Parent Functions Comparison Matrix
Algebra 1 Quick Reference
| Function |
Equation |
Visual Shape |
Domain |
Range |
Key Anchor |
| Linear |
\( f(x) = x \) |
Straight line |
All Reals |
All Reals |
\((0,0)\) Intercept |
| Quadratic |
\( f(x) = x^2 \) |
U-shape (parabola) |
All Reals |
\( y \ge 0 \) |
\((0,0)\) Vertex |
| Absolute Value |
\( f(x) = |
x |
\) |
V-shape (corner) |
All Reals |
| Square Root |
\( f(x) = \sqrt{x} \) |
Radical curve |
\( x \ge 0 \) |
\( y \ge 0 \) |
\((0,0)\) Endpoint |
Key Observation: Every single parent function here passes through \((0,0)\) and \((1,1)\)!
Worked Example 1
Identify From Key Features
Step-by-Step Analysis
Given Function Clues
Domain: \( x \ge 0 \) (non-negative numbers only)
Range: \( y \ge 0 \)
Key Points: Passes through \((0,0)\), \((1,1)\), and \((4,2)\).
Task: Identify which parent function this is and explain how the domain gives it away immediately.
Classroom Solution
1
Check the Domain: Linear, quadratic, and absolute value functions can all take negative \(x\)-values. Only square root requires \(x \ge 0\).
2
Test the Points: With \(f(x) = \sqrt{x}\):
\(\sqrt{0} = 0\), \(\sqrt{1} = 1\), and \(\sqrt{4} = 2\) ✓
3
Confirm Shape: Starts at \((0,0)\) and curves gently to the right, matching the radical function graph.
Answer: Square Root Parent Function, \( f(x) = \sqrt{x} \)
Pro-Tip: Check the domain first—it eliminates 3 out of 4 parent functions instantly!
Worked Example 2
Spot the Difference: \( x^2 \) vs. \( |x| \)
Parabola vs. V-Shape
Quadratic: \( f(x) = x^2 \)
Curved U
- Graph Shape: Smooth U-turn at the vertex \((0,0)\).
- How It Grows: Grows faster and faster! The curve gets steeper as \(x\) moves away from zero.
- Table Pattern: Outputs grow quadratically: \(1 \rightarrow 4 \rightarrow 9 \rightarrow 16\).
Non-constant rate of change (curves upward gently near 0, then shoots up).
Absolute Value: \( f(x) = |x| \)
Pointed V
- Graph Shape: Sharp pointed V with straight sides.
- How It Grows: Constant slope of \(+1\) on the right and \(-1\) on the left.
- Table Pattern: Outputs grow linearly: \(1 \rightarrow 2 \rightarrow 3 \rightarrow 4\).
Constant rate of change on each side (never curves).
Key Distinction: Both have domain All Reals and range \(y \ge 0\), but one is curved and one has straight sides!
Check for Understanding
Whole-Class Discussion
Question 1
Restricted Domain
Which parent function cannot accept negative numbers as input, and why?
\( f(x) = \sqrt{x} \) (Domain: \( x \ge 0 \))
Question 2
Comparing Ranges
Why can \(f(x) = x\) produce negative \(y\)-values, but \(f(x) = x^2\) cannot?
Squaring any real number gives \( \ge 0 \)
Question 3
Shared Coordinates
What two coordinate points are on all four parent function graphs?
\((0,0)\) and \((1,1)\)
Next Topic Preview: Vertical & Horizontal Shifts Algebra 1 Foundation Series
