Functions: Quadratic Functions Introduction

Explore quadratic functions in the forms y = ax² and y = ax² + c, generate function tables, identify vertex (0, c) and axis of symmetry x = 0, and compare linear vs. quadratic growth.

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Learning Guide: The Coordinate System

Learning Guide: Introduction to Quadratic Functions

In previous chapters, all the functions we explored were linear functions (

y = mx + b

). As we have seen, linear functions have the variable

x

to the first power (

x^1

), their rate of change (slope) is constant, and they always draw a straight line.

Now, we enter the world of polynomial functions — algebraic expressions built from whole-number powers of

x

(such as

x^1, x^2, x^3

). What determines whether a polynomial function is linear or non-linear is the highest exponent (degree) of the variable

x

Exponent equals 1 ( $x^1$ ): The polynomial is linear with a constant rate of change, producing a straight line.
Exponent greater than 1 ( $x^n$ with $n > 1$ ): Any polynomial with a power greater than $1$ (such as $x^2, x^3$ ) is non-linear! Because the rate of change is not constant, the graph is no longer a straight line, but bends into a curve.

When the highest exponent is exactly $2$ (

x^2

), this non-linear polynomial is called a quadratic function (from the Latin quadratus, meaning square). Its graph forms a smooth, symmetric U-shaped curve called a parabola.

📏 Linear Polynomial ( $x^1$ ) — Straight Line

Standard form:

y = mx + b

. The highest exponent of

x

is exactly

1

. The rate of change (slope

m

) is constant, so the graph is always a straight line.

🎢 Quadratic Polynomial ( $x^2$ , Degree $2$ )

Standard form:

y = ax^2 + c

. The highest exponent is greater than

1

(degree

2

). The rate of change is not constant, so the graph bends into a curved parabola.

🌍Where Do Quadratic Equations Come From? Real-World Examples

📐 Geometric Area (The Origin of "Quadratic")

The word quadratic comes from the Latin quadratus ("square"). When two lengths depend linearly on

x

, multiplying them can naturally produce an

x^2

term:

Square with side $x$ : $\text{Area} = x \cdot x = x^2$ . If you double the side length from $3$ to $6$ , the area quadruples from $9$ to $36$ ( $2^2 = 4$ )!
Rectangular garden: If length is $x + 4$ and width is $x$ , $\text{Area} = x(x + 4) = x^2 + 4x$ .
Circular pizza: $\text{Area} = \pi r^2$ .

🏀 Projectile Motion & Gravity

When an athlete kicks a soccer ball or shoots a basketball, gravity pulls it downward with constant acceleration. The ball rises, reaches a peak (the vertex!), and curves down in a parabolic arc:

h(t) = -5t^2 + 20t

📈Interactive Parabola Explorer

Adjust the sliders for

a

and

c

to observe how the parabola opens, stretches, and shifts.

y = x^2

Coefficient a

a = 1

Constant c (Vertical Shift)

c = 0

Opens upwards (Minimum at vertex)

Vertex:

(0, 0)

Axis of Symmetry:

x = 0

Standard Forms: $y = ax^2$ and $y = ax^2 + c$

Leading Coefficient ( $a$ ): Controls the orientation (direction of opening) and width. If

a > 0

, the parabola opens upwards (smiling) and the vertex is a minimum. If

a < 0

, it opens downwards (frowning) and the vertex is a maximum. A larger

|a|

produces a narrower/steeper curve.

Vertex of the Parabola

For functions of the form

y = ax^2 + c

, the turning point (vertex) is always located at

(0, c)

a > 0

, the vertex is the lowest point (minimum). If

a < 0

, the vertex is the highest point (maximum).

🔢Function Tables & Parabolic Symmetry

When generating a table for a quadratic function, choose symmetric inputs around the axis of symmetry

x = 0

(such as

x = -2, -1, 0, 1, 2

). Notice that opposite

x

-values give identical outputs:

f(-1) = f(1)

and

f(-2) = f(2)

$x$	$x^2$	$y = ax^2 + c$	$(x, y)$
-3	9	9	$(-3, 9)$
-2	4	4	$(-2, 4)$
-1	1	1	$(-1, 1)$
0	0	0	$(0, 0)$
1	1	1	$(1, 1)$
2	4	4	$(2, 4)$
3	9	9	$(3, 9)$

Visual Symmetry: Reflection Across the

y

-Axis (

x = 0

)

Every pair of opposite inputs (e.g.

x = -1

and

x = 1

) is at the exact same horizontal distance from the axis of symmetry (

x = 0

), producing identical

y

-values:

f(-x) = f(x)

Axis of Symmetry ( $x = 0$ ): The vertical line passing through the vertex. Because

(-x)^2 = x^2

, the curve on the left is a mirror image of the curve on the right:

f(-x) = f(x)

🚀Linear vs. Quadratic Growth Rates

Linear:

y = 3x

Quadratic:

y = x^2

$x$	$y = 3x$	Differences ( $\Delta y$ )	$y = x^2$	1st Difference ( $\Delta y$ )	2nd Difference
0	0	-	0	-	-
1	3	3	1	1	-
2	6	3	4	3	2
3	9	3	9	5	2
4	12	3	16	7	2
5	15	3	25	9	2

💡 What is $\Delta$ (Delta)? In mathematics, the Greek letter $\Delta$ stands for "change" or "difference" (here,

\Delta y

is the change in

y

from one row to the next: current

y

minus previous

y

).

8th Grade Pattern Rule (Difference of Differences):

Linear ( $y = 3x$ ): The differences ( $\Delta y$ ) are constant (always $+3$ ).
Quadratic ( $y = x^2$ ): The 1st differences ( $\Delta y$ ) keep growing ( $+1, +3, +5, +7, +9$ ). But notice the difference of the differences: $3 - 1 = \mathbf{2}$ , $5 - 3 = \mathbf{2}$ , $7 - 5 = \mathbf{2}$ . They are always constant ( $+2$ )!

A linear function changes at a constant rate (its first differences are always equal when the inputs increase by equal amounts). For a quadratic function, when the inputs increase by

1

each step, the second differences are constant and equal to

2a

. This is how we can identify a quadratic pattern from a table!

Learning Topics

Frequently Asked Questions

Why does squaring a negative number result in a positive value?

Because multiplying two negative numbers yields a positive product:

(-3)^2 = (-3) \times (-3) = +9

. This is why the parabola is symmetric across the y-axis:

f(-x) = a(-x)^2 + c = ax^2 + c = f(x)

Why is the axis of symmetry always

x = 0

for

y = ax^2 + c

Because there is no linear term (

bx

). The turning point occurs where

x = 0

, making the y-axis (

x = 0

) the exact fold line where both halves mirror each other.

How can you tell if the vertex is a minimum or maximum without graphing?

Look at the sign of coefficient

a

. If

a > 0

(positive), the parabola opens upwards like a cup, so the vertex is the lowest point (minimum). If

a < 0

(negative), it opens downwards like an arch, so the vertex is the highest point (maximum).

Why does a positive quadratic function eventually outgrow any linear function?

A linear function adds the same amount

m

with each step. A quadratic function with a positive leading coefficient has an increasingly large rate of change. For sufficiently large positive

x

, the

x^2

term grows faster than any linear term.