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Mathematics EN 9th grade: Quadratic function of the form $ f(x) = ax^2 $

1. What is a quadratic function (basic form)

A quadratic function is a function of the form:

$ f(x) = ax^2 $

Where:

$ a $ is a real number different from zero ($ a \neq 0 $)

$ x $ is the variable

$ ax^2 $ indicates that the highest degree term is the square of $ x $

📌 This is the simplest form of a quadratic function (without terms $ bx $or a constant).

2. Graph shape

The graph of$ f(x) = ax^2 $ is a parabola.

Main characteristics:

The vertex is at the origin $ (0,0) $

The axis of symmetry is the$ y $ axis

The shape depends on the sign of $ a $

👉 There are two cases:

✔ If $ a > 0 $:

The parabola opens upward

The minimum point is the vertex

✔ If $ a < 0 $:

The parabola opens downward

The maximum point is the vertex

Examples of graphs (description)

Case 1: $ f(x) = x^2 $ (ou seja, $ a = 1 $)#L# The parabola opens upward

It passes through points such as:

$ (-2, 4) $

$ (-1, 1) $

$ (0, 0) $

$ (1, 1) $

$ (2, 4) $

Case 2: $ f(x) = -x^2 $ (ou seja, $ a = -1 $)

The parabola opens downward

It passes through points such as:

$ (-2, -4) $

$ (-1, -1) $

$ (0, 0) $

$ (1, -1) $

$ (2, -4) $

4. Value table (example)

For $ f(x) = 2x^2 $:

When $ x = -2 $, $ f(x) = 2 \cdot (-2)^2 = 8 $

When $ x = -1 $, $ f(x) = 2 \cdot (-1)^2 = 2 $

When $ x = 0 $, $ f(x) = 0 $

When $ x = 1 $, $ f(x) = 2 \cdot 1^2 = 2 $

When $ x = 2 $, $ f(x) = 2 \cdot 2^2 = 8 $

✔ Symmetry is observed with respect to the$ y $ axis

5. Important interpretation

The value of $ x^2 $ is always positive or zero

The sign of $ a $ determines the opening of the parabola

The larger the absolute value of $ a $, the narrower the parabola

The smaller the absolute value of $ a $, the wider the parabola

6. Key points for the exam

✔ The function $ f(x) = ax^2 $ forms a parabola

✔ The vertex is at $ (0,0) $

✔ The axis of symmetry is the $ y $ axis

✔The sign of $ a $ determines whether it opens upward or downward

✔ The graph is symmetric with respect to the $ y $ axis.

Did you know?

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Explore more about What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a
Examples:
\( |5| = 5 $
$ |-5| = 5 $
Simétrico
The Simétrico of a number is:
$ -a $
Examples:
$ 3 \rightarrow -3 $
$ -7 \rightarrow 7 $
Adição e subtração
Addition with the same sign
Add the values and keep the sign
Examples:
$ 3 + 5 = 8 $
$ -4 + (-6) = -10 $
Soma com sinais contrários
Subtract the values and take the sign of the number with the greatest absolute valueNL#
Examples:
$ 7 + (-3) = 4 $
$ -8 + 5 = -3 $
Soma de números simétricos
The result is always zero
$ a + (-a) = 0 $
Examples:
$ 5 + (-5) = 0 $
Subtração (diferença)
It is transformed into the sum of the opposite
$ a - b = a + (-b) $
Examples:
$ 7 - 3 = 7 + (-3) = 4 $
$ 5 - (-2) = 5 + 2 = 7 $
Divisão de números racionais
Main rule
Multiply by the reciprocal
$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $
Inverso de um número racional
Swap the numerator and denominator
$ \text{Inverso de } \frac{a}{b} = \frac{b}{a}, \quad a \neq 0 $
Examples:
$ \frac{3}{4} \rightarrow \frac{4}{3} $
$ 2 = \frac{2}{1} \rightarrow \frac{1}{2} $
Important:
$ 0 \text{ não tem inverso} $
Examples de divisão
$ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} $
TIPS
✔ Same signs → add and keep the sign
✔ Different signs → subtract and take the sign of the number with the greatest absolute value
✔ Subtraction → convert to addition
✔ Division → multiply by the reciprocal
✔ Número + simétrico: $ a + (-a) = 0 $
✔ Valor absoluto: $ |a| \geq 0 $
..

What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a < 0 $
Examples:
$ |5| = 5 $
$ |-5| = 5 $
Simétrico
The ..know more

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- Formas tónicas: used alone or for emphasis (mim, ti, ele, nós).
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Pronome pessoal em adjacência verbal
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Dei-lho
Pronome pessoal “se”
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Pronome demonstrativo
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Pronome possessivo
Indicates possession: meu, teu, seu, nossa, vossos.
Pronome indefinido
Indicates vague quantity or identity: alguém, ninguém, tudo, algum, certo.
Pronome relativo
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Word that replaces or accompanies a noun, avoiding repetition and indicating person, possession, quantity, or relation.
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Subclasses de pronomes
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