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Chapter 8: Problem 2

Fill in the blanks. The graph of \(f(x)=x^{2}\) is a cuplike shape called a _____.

Short Answer

Expert verified
The graph of \(f(x)=x^{2}\) is a cup-like shape called a parabola.

Step by step solution

01

Understanding the Graph of a Quadratic Function

The function given is a quadratic function: \(f(x) = x^2\). Quadratic functions that have the form \(f(x) = ax^2 + bx + c\) typically produce a specific type of graph.
02

Identifying Common Graph Shape

The shape produced by a quadratic function \(f(x) = x^2\) when graphed is known to form a 'u-shape.' This is a common characteristic curve associated with a particular shape in mathematics.
03

Naming the Shape

The 'u-shaped' curve produced by the function \(f(x) = x^2\) is specifically called a 'parabola.' In this case, since the coefficient of \(x^2\) is positive, it opens upwards, solidifying the 'cup-like' appearance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Parabolas
A parabola is a fundamental concept in mathematics, particularly in the study of quadratic functions. When you graph a basic quadratic function like \(f(x) = x^2\), you will see a smooth, symmetrical curve. This curve is called a parabola.
Parabolas have several essential characteristics:
  • They are symmetrical around a vertical line called the axis of symmetry.
  • They have a highest or lowest point, called the vertex, where the direction of the curve changes.
  • The opening direction of a parabola is determined by the sign of the coefficient in front of the \(x^2\) term; if it's positive, the parabola opens upwards; if negative, it opens downwards.
Understanding these properties helps you recognize and draw parabolas easily on a graph.
The Graph of a Quadratic Function
The quadratic function \(f(x) = ax^2 + bx + c\) creates a graph that is a parabola. The graph of this function is an essential component in understanding algebra and mathematics as a whole.
The graph has some distinct features:
  • The axis of symmetry runs vertically through the vertex of the parabola at \(x = -\frac{b}{2a}\).
  • The y-intercept is the point where the graph crosses the y-axis, located at \(c\).
  • The x-intercepts, if they exist, are the points where the parabola crosses the x-axis and can be found using the quadratic formula.
Knowing how to graph a quadratic function is a valuable skill that develops a deeper understanding of mathematical concepts.
Significance of the U-Shaped Curve
The u-shaped curve is a distinct characteristic of quadratic functions. When you visualize the function \(f(x) = x^2\), the image that comes to mind is the classic u-shape.
This shape is not just a simple curve; it has critical implications:
  • A positive a-value in the quadratic function means the u-shape opens upwards like a cup, representing a minimum point at the vertex.
  • A negative a-value flips the u-shape downwards like a dome, indicating a maximum point at the vertex.
  • The steepness of the curve is controlled by the coefficient \(a\). A larger absolute value of \(a\) results in a narrower parabola.
Understanding this u-shaped curve helps in various fields such as physics and engineering, where parabolic motion and characteristics appear frequently.

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Most popular questions from this chapter

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{3}{s-2}+\frac{s-14}{2 s^{2}-3 s-2}-\frac{4}{2 s+1}=0$$

Perform the operations and simplify, if possible. See Example \(8 .\) $$\frac{8}{9 y^{2}}+\frac{1}{6 y^{4}}$$

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through \((1,7)\) and \((-2,1)\)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope \(2, y\) -intercept \((0,11)\)

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{34}{x^{2}}+\frac{13}{20 x}=\frac{3}{2 x}$$
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