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Chapter 3: Problem 2

Fill in the blank. If \(a<0,\) the graph of \(f(x)=a(x-h)^{2}+k\) opens __________

Short Answer

Expert verified
The graph of f(x) opens downwards.

Step by step solution

01

Understanding the vertex form of a quadratic function

The quadratic function is given in the vertex form: Note that the equation is f(x)=a(x-h)^{2}+k. The values of h and k will determine the vertex of the parabola, but the sign of a will determine if the parabola opens upwards or downwards.
02

Analyzing the coefficient a

The coefficient a in the quadratic function determines the direction of the parabola. If a is positive, the parabola opens upwards. If a is negative, the parabola opens downwards.
03

Applying given condition a < 0

We are given that a < 0. According to the properties of the quadratic function, if a is less than 0, the graph of the function will open 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 U-shaped curve that is the graph of a quadratic function. The general form of a quadratic function is expressed in various forms, with one being the vertex form: \[ f(x) = a(x-h)^2 + k \] In this form, \(h, k\) represents the vertex of the parabola, which is its highest or lowest point. The parabola can open either upwards or downwards depending on the value of the coefficient a.
  • If the parabola opens upwards, it has a minimum point at its vertex.
  • If it opens downwards, it has a maximum point at its vertex.
Understanding how the values of a, h, and k affect the shape and position of the parabola is key to mastering quadratic functions.
coefficient a
The coefficient a in the vertex form of a quadratic function plays a crucial role in determining the shape and direction of a parabola. The general vertex form is given as: \[ f(x) = a(x-h)^2 + k \]
  • If a is positive, the parabola opens upwards.
  • If a is negative, the parabola opens downwards.
The value of \(a\) not only affects the direction in which the parabola opens but also influences its width.
  • A larger absolute value of a (e.g., \( |a| > 1 \)) makes the parabola narrower.
  • A smaller absolute value of a (e.g., \( |a| < 1 \)) makes the parabola wider.
Thus, \(a\) is fundamental to understanding the graph's orientation and shape.
graph direction
The direction of the graph of a quadratic function is determined by the coefficient a in its equation form \[ f(x) = a(x-h)^2 + k \] To determine the graph's direction, you simply look at the sign of the coefficient a.
  • If \(a > 0\), the graph opens upwards.
  • If \(a < 0\), the graph opens downwards.
This direction is important as it indicates whether the vertex is a minimum or maximum point on the graph. In summary, by examining the sign of the coefficient a, you can quickly determine the general behavior of the quadratic function鈥檚 graph.

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