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Chapter 3: Problem 7

Given \(f(x)=a(x-h)^{2}+k,\) if \(a>0,\) then the minimum value of \(f\) is _____.

Short Answer

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k

Step by step solution

01

Identify the form of the function

The given function is in the vertex form of a quadratic equation: \[ f(x) = a(x - h)^2 + k \] where \(a\), \(h\), and \(k\) are constants.
02

Recognize the importance of the parameter 'a'

Since \(a > 0\), the parabola opens upwards. This means the vertex of the parabola represents the minimum point of the function.
03

Identify the vertex

The vertex of the quadratic function \(f(x) = a(x - h)^2 + k\) is the point \((h, k)\).
04

Determine the minimum value

At the vertex \((h, k)\), the value of the function \(f(x)\) is \(k\). Therefore, the minimum value of the function is \(k\).

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Key Concepts

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

Vertex Form
The vertex form of a quadratic function is a special way of expressing quadratics. It's given by \[ f(x) = a(x - h)^2 + k \] Here,
  • \(a\) indicates the direction and width of the parabola. If \(a > 0\), it means the parabola opens upwards.
  • \(h\) and \(k\) help to determine the vertex of the parabola. The vertex is the point \((h, k)\), the turning point of the graph.
Understanding the vertex form allows you to easily identify the vertex and whether the parabola opens upwards or downwards based on \(a\). This form simplifies finding the maximum or minimum values of quadratic functions.
Parabola
A parabola is the graph of a quadratic function. When we talk about a parabola in the context of quadratics, it takes the shape of a symmetrical 'U'.
If \(a > 0\), the parabola opens upwards, making the vertex the minimum point. Conversely, if \(a < 0\), the parabola opens downwards with the vertex being the maximum point.
A parabola is defined by its vertex, axis of symmetry, and direction. The vertex form \( f(x) = a(x - h)^2 + k \) makes determining these characteristics straightforward. The vertex \((h, k)\) is at the core, and the line \(x = h\) is the axis of symmetry.
Minimum Value
In the context of quadratic functions, the minimum value represents the lowest point on the graph of the parabola. For a function given by \( f(x) = a(x - h)^2 + k \), if \(a > 0\), the parabola opens upwards.
  • This implies the vertex is the minimum point.
  • Since the vertex is at \((h, k)\), the minimum value of the function is simply \(k\).
So, when you see a quadratic function in vertex form and you need to find the minimum value, you just look at the 鈥榢鈥� part of the equation. This value is the lowest the function will reach.

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Most popular questions from this chapter

Determine if the statement is true or false. a. Use the quadratic formula to solve \(x^{2}-7 x+5=0\) b. Write \(x^{2}-7 x+5\) as a product of linear factors.

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