Problem 24 - Algebra Form and Function

Question

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=10-(x-2)^{2} $$

Step-by-Step Solution

Answer

Answer: The range of the function \(y = 10 - (x-2)^2\) is \(y \leq 10\).

Step 1: Analyze the given function

We are given the function \(y = 10 - (x-2)^2\) and want to find its range.

Step 2: Compare with the similar function

Compare the given function with the function that has a known range: \(y = 9 - (x - 2)^2\). Notice that the only difference between the two functions is the constant term (10 in the given function and 9 in the similar function). This means that the given function is simply a vertical translation of the similar function (up one unit).

Step 3: Determine the range of the translated function

Since the given function is just a vertical translation of the similar function with a known range, the range of the given function will be similarly affected. So, if the range of the similar function is \(y \leq 9\), then the range of the given function will be \(y \leq 10\) (up one unit).

Step 4: Write the final answer

The range of the function \(y=10-(x-2)^2\) is \(y \leq 10\).

Key Concepts

Vertex Form of a Quadratic Function

In mathematics, especially within algebra, the vertex form of a quadratic function is a particular way of expressing a quadratic equation. A quadratic equation in its standard form is typically represented as \( ax^2 + bx + c \). However, the vertex form provides a different perspective.

• It is given by \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
• Think of \( a \) as the "stretch" factor that influences the width and direction of the parabola. A positive \( a \) value means the parabola opens upwards, while a negative value points it downwards.
• The parameters \( h \) and \( k \) help identify the position of the vertex.

Recognizing a quadratic in vertex form is very useful because it allows us to immediately see the vertex of the parabola. For example, in the function \( y = 9 - (x-2)^2 \), it's clear that the vertex is at \((2, 9)\). This insight can greatly aid in graphing this function.

Vertical Translation

Vertical translation in the context of graphing functions is a simple and intuitive concept. It refers to the movement of a graph up or down in a Cartesian plane.

• A function \( f(x) \) translated vertically by \( c \) units looks like \( y = f(x) + c \).
• If \( c \) is positive, the whole graph shifts upward; if \( c \) is negative, it shifts downward.

Consider our specific example where the function \( y = 9 - (x-2)^2 \) changes to \( y = 10 - (x-2)^2 \). The only alteration is a shift up by 1 unit. This translation alters the range of the function since the highest point (the vertex) increases by 1, adjusting the maximum possible value of \( y \). This vertical shift results in the new range being \( y \leq 10 \). Understanding vertical translation is crucial when analyzing how modifications to a function affect its graph.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial. This method is essential when working with quadratic equations, especially when converting them into vertex form.

• Begins with the general quadratic form \( ax^2 + bx + c \).
• Divide the linear term coefficient \( b \) by 2, and square the result. This value is then added and subtracted within the expression to maintain equality.
• Reorganize the expression to form \((x+d)^2 + e\), where \( d \) and \( e \) are constants.

Applying this technique simplifies the process of finding important features of a quadratic, like its vertex. For example, starting with the expression \( x^2 + 4x \), completing the square would transform it into \( (x+2)^2 - 4 \). Recognizing this converts the standard form into something much more telling about the graph of the function.