Goal of This Unit:

By the end of this unit, students should be able to:

• Understand the general form of quadratic and cubic functions.

• Identify key features such as intercepts, turning points, and end behavior.

• Graph quadratic and cubic functions accurately.

• Recognize how the degree of a function affects its shape and behavior.

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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.

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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.

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Cubic Functions (Third Degree)

Third-degree functions, or cubic functions, are polynomials where the highest exponent on the variable is 3. The general form of a cubic function is f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants, and a is not zero.

Unlike quadratic functions, the graph of a cubic function can have a much more complex shape. Simple cubic functions, like f(x) = x³, create a smooth curve that moves from the bottom left of the graph to the top right if a is positive, or from the top left to the bottom right if a is negative.

More complicated cubic functions can have up to two turning points — places where the graph changes direction from increasing to decreasing or vice versa. Another unique feature of cubic graphs is the inflection point, where the curve changes the way it bends, transitioning from concave up to concave down or the other way around.

The behavior of a cubic function at the far ends of the graph — what we call its end behavior — is determined by the leading coefficient a. If a is positive, the graph will rise on the right and fall on the left. If a is negative, it will rise on the left and fall on the right.

To graph a cubic function, we calculate several points by choosing different x-values and computing the corresponding f(x) values. We plot these points carefully, noting any intercepts, turning points, and the general shape of the curve.

Cubic functions are also very common in real-life contexts, particularly in problems involving volume, population growth models, and certain physics equations.

By understanding both quadratic and cubic functions, you will be well prepared to analyze more complex behaviors in functions and ready to move on to learning about how these behaviors change, which is the focus of derivatives in calculus.

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video graphing cubic functions

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