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Chapter 3: Problem 1

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

Short Answer

Expert verified
Upwards

Step by step solution

01

- Recognize the form of the quadratic function

Identify the standard form of the quadratic function given in the problem, which is \(f(x)=a(x-h)^{2}+k\).
02

- Determine the value of 'a'

Notice that the problem specifies that \(a>0\). This condition on 'a' will determine the direction in which the graph opens.
03

- Analyze the role of 'a'

Understand that for a quadratic function in the form \(f(x)=a(x-h)^{2}+k\), if \(a>0\), the parabola opens upwards. Conversely, if \(a<0\), it opens downwards. Here, since \(a>0\), the parabola will open upwards.
04

- Fill in the blank

Based on the analysis, fill in the blank with the appropriate direction: 'upwards.'

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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 you get when you graph a quadratic function. Imagine the path of a ball thrown into the air.
This curve is symmetric, meaning one side mirrors the other. The highest or lowest point on a parabola is known as the vertex.
When you graph a quadratic function, the result is a parabola.
Understanding parabolas is essential for solving quadratic equations and analyzing their properties.
vertex form
The vertex form of a quadratic function is a special way to write the equation, making it easier to see and understand. It looks like this: \(f(x) = a(x-h)^{2} + k\).
Here's what each part means:
  • \(a\): This number tells us how 'wide' or 'narrow' the parabola is and its direction of opening.
  • \(h\): This value moves the parabola left or right along the x-axis.
  • \(k\): This moves the parabola up or down along the y-axis.
The point \((h, k)\) is the vertex of the parabola. Knowing the vertex form helps in quickly identifying critical aspects of the graph.
direction of opening
The direction of opening of a parabola depends on the value of 'a' in the vertex form equation \(f(x)=a(x-h)^{2}+k\).
Here's what you need to know:
  • If \(a > 0\), the parabola opens upwards, creating a U-shape like a cup.
  • If \(a < 0\), the parabola opens downwards, creating an inverted U-shape like an upside-down cup.
Understanding this concept is crucial in graphing and solving quadratic equations, as it gives immediate insights into the graph's behavior.

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