Relations vs functions

Quadratic expressions

Quadratic functions

Quadratic equations

MYP Standard Mathematics Quadratic Functions Notes

Key Features Of Any Quadratic Graph

When you sketch or analyze a quadratic, you nearly always focus on three main characteristics:three main characteristics:

Intercepts (where the graph crosses the axes)

Vertex (the turning point)

Axis of symmetry (the vertical line through the vertex)

Definition

x-Intercepts (zeros)

The $x$-coordinates where the graph crosses the $x$-axis (where $y=0$). They are also called the zeros (or roots) of the function.

Definition

y-Intercept

The point where the graph crosses the $y$-axis (where $x=0$).

Definition

Vertex

The turning point of the parabola, with coordinates $(x_v, y_v)$.

Definition

Axis of symmetry

The vertical line that divides the parabola into two mirror-image halves. Its equation is $x=x_v$.

How The Parameters $a$, $b$, And $c$ Affect The Parabola

In standard form, $$y=ax^2+bx+c,\quad a\neq 0$$

$a$ controls opening and steepness:

If $a>0$, the parabola opens upward (minimum point).
If $a<0$, the parabola opens downward (maximum point).
The larger $|a|$ is, the narrower (steeper) the parabola; if $0<|a|<1$, it is wider.

$c$ is the y-intercept (because when $x=0$, $y=c$).

$b$ influences where the vertex sits horizontally, and therefore the axis of symmetry.

Note

The parameter $b$ does not have a simple "one-step" interpretation like $c$, but it is crucial because it shifts the axis of symmetry and vertex left or right.

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Note

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Tip

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In the previous article, we covered quadratic expressions.
Now, we can talk about quadratic functions.
A "U-shaped" (or upside-down U-shaped) relationships such as projectiles, arches, and optimization problems.

Definition

Quadratic function

A function whose graph is a parabola and can be written in the form $y=ax^2+bx+c$ where $a\neq 0$.

Key Features Of Any Quadratic Graph

When you sketch or analyze a quadratic, you nearly always focus on three main characteristics:three main characteristics:

Intercepts (where the graph crosses the axes)
Vertex (the turning point)
Axis of symmetry (the vertical line through the vertex)

Definition

x-Intercepts (zeros)

The $x$-coordinates where the graph crosses the $x$-axis (where $y=0$). They are also called the zeros (or roots) of the function.

Definition

y-Intercept

The point where the graph crosses the $y$-axis (where $x=0$).

Definition

Vertex

The turning point of the parabola, with coordinates $(x_v, y_v)$.

Definition

Axis of symmetry

The vertical line that divides the parabola into two mirror-image halves. Its equation is $x=x_v$.

How The Parameters $a$, $b$, And $c$ Affect The Parabola

In standard form, $$y=ax^2+bx+c,\quad a\neq 0$$
$a$ controls opening and steepness:
1. If $a>0$, the parabola opens upward (minimum point).
2. If $a<0$, the parabola opens downward (maximum point).
3. The larger $|a|$ is, the narrower (steeper) the parabola; if $0<|a|<1$, it is wider.
$c$ is the y-intercept (because when $x=0$, $y=c$).
$b$ influences where the vertex sits horizontally, and therefore the axis of symmetry.

Note

The parameter $b$ does not have a simple "one-step" interpretation like $c$, but it is crucial because it shifts the axis of symmetry and vertex left or right.

Intercepts: Fast Ways To Find Them

Finding The y-Intercept

Set $x=0$:
So the y-intercept is $(0,c)$.

MYP Standard Mathematics Quadratic Functions Notes

Key Features Of Any Quadratic Graph

How The Parameters $a$, $b$, And $c$ Affect The Parabola

Intercepts: Fast Ways To Find Them

Finding The y-Intercept

Finding x-Intercepts By Solving $y=0$

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Key Features Of Any Quadratic Graph

How The Parameters $a$, $b$, And $c$ Affect The Parabola

Intercepts: Fast Ways To Find Them

Finding The y-Intercept

Finding x-Intercepts By Solving $y=0$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

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Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

MYP Standard Mathematics Quadratic Functions Notes

Number and Algebra Foundations7 subtopics

Coordinate Geometry and Linear Modelling5 subtopics

Functions and Quadratic Relationships4 subtopics

Sets, Logic and Probability4 subtopics

Geometry of Shapes, Surface Area and Volume3 subtopics

Statistical Measures and Data Representation3 subtopics

Right Triangles and Circle Geometry3 subtopics

Sequences, Formulae and Proportional Reasoning3 subtopics

Sampling Methods and Bivariate Data2 subtopics

Key Features Of Any Quadratic Graph

How The Parameters $a$, $b$, And $c$ Affect The Parabola

Intercepts: Fast Ways To Find Them

Finding The y-Intercept

Finding x-Intercepts By Solving $y=0$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Key Features Of Any Quadratic Graph

How The Parameters $a$, $b$, And $c$ Affect The Parabola

Intercepts: Fast Ways To Find Them

Finding The y-Intercept

Finding x-Intercepts By Solving $y=0$

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account