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Chapter 1: Problem 18

Use your graphing calculator to graph \(y=a x^{2}\) for \(a=\frac{1}{5}, 1\), and 5 , then again for \(a=-\frac{1}{5},-1\), and \(-5\). Copy all six graphs onto a single coordinate system and label each one. Explain how a negative value of \(a\) affects the parabola.

Short Answer

Expert verified
A negative value of \( a \) causes the parabola to open downward.

Step by step solution

01

Understand the Quadratic Function

We are working with the quadratic function \( y = ax^2 \), where the coefficient \( a \) determines the shape and orientation of the parabola. We will graph this function for different values of \( a \).
02

Graphing Positive Values of 'a'

Use your calculator to graph the function for positive values of \( a \): - For \( a = \frac{1}{5} \), the graph is a wide upward-opening parabola.- For \( a = 1 \), the graph is a standard upward-opening parabola.- For \( a = 5 \), the graph is a narrow upward-opening parabola.Plot these three graphs on the same coordinate system, labeling each one accordingly.
03

Graphing Negative Values of 'a'

Now graph the function for negative values of \( a \):- For \( a = -\frac{1}{5} \), the graph is a wide downward-opening parabola.- For \( a = -1 \), the graph is a standard downward-opening parabola.- For \( a = -5 \), the graph is a narrow downward-opening parabola.Plot these three graphs on the same coordinate system, labeling each one accordingly.
04

Analyze the Effect of Negative 'a'

When \( a \) is negative, the parabola opens downward instead of upward. The absolute value of \( a \) determines how wide or narrow the parabola appears, without affecting the direction of the opening.
05

Copy All Graphs onto a Single Coordinate System

Ensure all six parabolas are plotted on the same coordinate system. Label each parabola with its respective value of \( a \). This visual representation helps compare the effects of different \( a \) values, both positive and negative.

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Key Concepts

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

Parabolas
Parabolas are U-shaped curves that are graphically represented by quadratic functions. When graphed, the basic form of a parabola is expressed as \( y = ax^2 \). The value of \( a \) influences certain characteristics of the parabola, such as its width and direction of opening. In general:
  • If \( a > 0 \), the parabola opens upward, resembling a smile.
  • If \( a < 0 \), the parabola opens downward, resembling a frown.
Understanding the nature of parabolas is fundamental because these shapes frequently appear in mathematical contexts, from physics problems involving projectile motion to engineering designs. When working with quadratics, it's essential to focus on how their specific structure influences their graphical representation.
Quadratic Function Characteristics
Quadratic functions, typically expressed in the form \( y = ax^2 + bx + c \), have distinct characteristics. However, examining the simpler form \( y = ax^2 \) sheds light on fundamental attributes:
  • Vertex: This is the highest or lowest point of the parabola, depending on its orientation. For \( y = ax^2 \), the vertex is located at the origin, \( (0,0) \).
  • Axis of Symmetry: Parabolas are symmetrical about a vertical line called the axis of symmetry. In our case, it's the y-axis, or \( x = 0 \).
  • Direction of Opening: As mentioned, the sign of \( a \) determines whether it opens upwards or downwards.
  • Width: The absolute value of \( a \) affects the width. Smaller values result in wider parabolas, while larger values cause narrower ones.
Recognizing these characteristics allows for the prediction and understanding of the behavior of any quadratic function graphically.
Effects of Coefficient on Graph Shape
The coefficient \( a \) in the equation \( y = ax^2 \) holds significant sway over the graph's appearance. Here鈥檚 a breakdown of how \( a \) affects the parabola:
  • Positive & Negative Values of \( a \): As previously noted, positive values of \( a \) yield upwards-opening parabolas, while negative values flip the parabola to open downwards.
  • Magnitude of \( a \): The width of the parabola is contingent on \( |a| \), the absolute value of \( a \). The smaller \( |a| \) is, the wider the parabola becomes; conversely, as \( |a| \) increases, the parabola becomes narrower.
Consider graphing different \( a \) values to see these effects. For example, with \( a = \frac{1}{5} \) we see a broad upward-facing curve, while \( a = 5 \) results in a steeper, more narrow formation. Similarly, negative counterparts like \( a = -1 \) or \( a = -5 \) invert the parabola downwards while maintaining width corresponding to \( |a| \). Observing these attributes across multiple values of \( a \) can offer a comprehensive view of quadratic behaviors and their visual outcomes.

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Most popular questions from this chapter

Draw \(45^{\circ}\) and \(-45^{\circ}\) in standard position and then show that \(\cos \left(-45^{\circ}\right)=\) \(\cos 45^{\circ}\).

Multiply. \((\sin \theta+4)(\sin \theta+3)\)

Show that each of the following statements is an identity by transforming the left side of each one into the right side. \(\sin \theta \cot \theta=\cos \theta\)

Write each of the following in terms of \(\sin \theta\) only: \(\sec \theta\)

\(\cos \theta(\sec \theta-\cos \theta)=\sin ^{2} \theta\)
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