Intro to Functions
Welcome to The Math Forge complete beginner’s guide to calculus—designed for you, the curious mind with little to none prior experience in calculus (or even advanced algebra). Whether you're a student, a lifelong learner, or someone who’s always felt math wasn’t for them, this course will walk you—step by step—into one of the most powerful ideas humans have ever created.
Calculus is the math of change and accumulation. It helps us understand how fast things move, how populations grow, how planets orbit, and even how your coffee cools over time. But don’t worry—before we touch a formula, we’ll build the intuitive foundation so it all makes sense.
Unit 2 Complex Functions
Understanding Functions, Variables, and Graphs
Goal of This Unit:
By the end of this unit, students should be able to:
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Understand the general form of quadratic and cubic functions.
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Identify key features such as intercepts, turning points, and end behavior.
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Graph quadratic and cubic functions accurately.
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Recognize how the degree of a function affects its shape and behavior.
Why This Matters:
Before you learn calculus, you need to understand how math communicates. Math is a language—and if we want to talk about motion, change, and patterns, we first need the vocabulary.
Quadratic Functions (Second Degree)
Second-degree functions, also known as quadratic functions, are polynomials where the highest exponent on the variable is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants, and a is not zero.
The graph of a quadratic function is a smooth curve called a parabola. This parabola either opens upwards or downwards depending on the value of a. If a is positive, the parabola opens upwards like a U, this is known as a positive concavity. If a is negative, the parabola opens downwards like an upside-down U, know as a negative concavity.
An essential feature of every quadratic function is the vertex. The vertex is the highest point if the concavity is negative, or the lowest point if the concavity is positive. The vertex also lies on the axis of symmetry, which is an invisible vertical line that splits the parabola into two symmetrical halves.
To graph a quadratic function, we usually start by finding the vertex. This can be done using the formula for the x-coordinate of the vertex: x = -b/(2a). After finding the vertex, we identify the y-intercept by calculating f(0), and look for any x-intercepts by solving the equation f(x) = 0, often using factoring or the quadratic formula.
Once these key points are identified, we plot them and then draw a smooth curve through them to complete the graph.
What Is a Function?
A function is a rule that tells us how to turn input into output.
Example: f(x)=2x+1
This means: Take a number (like 3)
Multiply it by 2 → 6
Add 1 → 7
So: f(3)=7
Think of it like a vending machine:
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You put in a number (input)
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The machine follows a rule
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You get a result (output)
Graphing a Function
Let’s graph f(x)=2x+1