Question

$$f(x)=0.3 x^{2}-4$$ (GRAPH NOT COPY)

   $$f(x)=0.3 x^{2}-4$$
(GRAPH NOT COPY)

College Algebra with Modeling and Visualization

College Algebra with Modeling and Visualization

Gary K. Rockswold 4th Edition

Chapter 1, Problem 50

Instant Answer

Solved by Expert Melissa Barry

06/19/2021

Step 1

3 x^{2}-4$. This is a quadratic function, which means it will form a parabola when graphed. Show more…

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$$f(x)=0.3 x^{2}-4$$ (GRAPH NOT COPY)

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Key Concepts

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point of the graph, depending on whether it opens downward or upward. It can be determined by the formula (-b/(2a), f(-b/(2a))) when the quadratic is in standard form. Knowing the vertex is crucial for graphing the parabola and understanding its symmetry and optimal value, which are important concepts in both algebra and calculus.

Quadratic Function

A quadratic function is a type of polynomial function of degree two, typically written in the form f(x) = ax^2 + bx + c. Its graph is a parabola, which can open upward if the coefficient a is positive or downward if a is negative. Understanding quadratic functions is fundamental in algebra and precalculus, as they model various phenomena and provide a basis for further study in calculus and other higher-level math topics.

Standard Form

The standard form of a quadratic function is expressed as f(x) = ax^2 + bx + c. This form is useful for identifying key characteristics of the function such as the coefficient that influences the stretching or compressing of the graph, the linear coefficient that affects the axis of symmetry, and the constant that determines the vertical position of the parabola. In the example provided, the function is already in standard form, making it easier to analyze its properties.

Graph Transformations

Graph transformations involve shifting, stretching, or compressing the basic graph of a function to obtain the graph of a more complex function. For quadratic functions, the coefficient 'a' determines vertical stretching (if |a| > 1) or compression (if 0 < |a| < 1), and the constant term 'c' results in a vertical translation. These transformations are key to understanding how changes in the function’s formula affect the shape and position of its graph.

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