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Chapter 3: Problem 5

Given \(f(x)=a(x-h)^{2}+k,\) if \(a<0,\) then the parabola opens ______.

Short Answer

Expert verified
The parabola opens downwards if a < 0.

Step by step solution

01

- Identify the Standard Form of a Parabola

The function given is in the form of a standard parabola, which is written as: f(x) = a(x-h)^2 + k. This form allows us to easily identify key features such as the direction in which the parabola opens.
02

- Analyze the Coefficient 'a'

The coefficient 'a' determines whether the parabola opens upwards or downwards. If 'a' is positive (a > 0), the parabola opens upwards. If 'a' is negative (a < 0), the parabola opens downwards.
03

- Apply the Given Condition

In the problem, it is given that a < 0. According to the previous step where the behavior of 'a' was analyzed, if 'a' is less than zero, the parabola opens downwards.

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Key Concepts

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

parabola
A parabola is a symmetrical, curved shape often seen in quadratic functions. It represents the graph of a quadratic equation and can open either upwards or downwards. Important features of a parabola include its vertex, which is its highest or lowest point, and its axis of symmetry, a vertical line that divides the parabola into two mirror-image halves. This curve is not just a theoretical concept; you'll find it in many real-life scenarios such as the path of a projectile or the design of satellite dishes.
standard form of a parabola
The standard form of a parabola is expressed as: \[ f(x) = a(x-h)^2 + k \]. In this equation,
  • 'a' controls the width and direction of the parabola,
  • 'h' shifts the parabola left or right, and
  • 'k' moves it up or down.
Understanding this form makes it easier to analyze the parabola's properties. For instance, when 'a' is positive, the parabola opens upwards, and when 'a' is negative, it opens downwards. The vertex of the parabola is at the point \( (h, k) \).
coefficient analysis
Analyzing the coefficient 'a' in the standard form of a parabola \[ f(x) = a(x-h)^2 + k \], is crucial for understanding its shape. Here鈥檚 how you can interpret 'a':
  • If \( a > 0 \), the parabola opens upwards. This means the vertex is the lowest point.
  • If \( a < 0 \), the parabola opens downwards, making the vertex the highest point.
  • The absolute value of 'a' affects the width; smaller values make the parabola wider, while larger values make it narrower.
By analyzing 'a', you can quickly determine key attributes of the parabola without needing to plot numerous points.
direction of parabola opening
The direction in which a parabola opens is determined by the sign of the coefficient 'a' in the standard quadratic function \[f(x) = a(x-h)^2 + k\].
  • When \( a > 0 \), the parabola opens upwards. The arms of the curve extend infinitely upwards, depicting a 'U'-shape.
  • When \( a < 0 \), the parabola opens downwards. The arms of the curve extend infinitely downwards, forming an 'n'-shape.
Understanding the direction of the parabola is essential for graphing and solving quadratic equations. As given in the original exercise, if \( a < 0 \), the parabola will open downwards.

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Most popular questions from this chapter

Determine if the statement is true or false. Given \(f(x)=2 i x^{4}-(3+6 i) x^{3}+5 x^{2}+7,\) if \(a+b i\) is a zero of \(f(x)\), then \(a-b i\) must also be a zero.

Write the domain of the function in interval notation. $$ s(x)=\frac{1}{\sqrt{4 x^{2}+7 x-2}} $$

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \left|x^{2}-4\right|<5 $$

The number of adults in U.S. prisons and jails for the years \(1980-2008\) is shown in the graph. (Source: U.S. Department of Justice, www.justice.gov) The variable \(t\) represents the number of years since 1980 . The function defined by \(P(t)=-0.091 t^{3}+3.48 t^{2}+15.4 t+335\) represents the number of adults in prison \(P(t)\) (in thousands). The function defined by \(J(t)=23.0 t+159\) represents the number of adults in jail \(J(t)\) (in thousands). a. Write the function defined by \(N(t)=(P+J)(t)\) and interpret its meaning in context. b. Write the function defined by \(R(t)=\left(\frac{J}{N}\right)(t)\) and interpret its meaning in the context of this problem. c. Evaluate \(R(25)\) and interpret its meaning in context. Round to 3 decimal places.

Explain how the solution set to the inequality \(f(x)<0\) is related to the graph of \(y=f(x)\).
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