Problem 5 - Beginning and Intermediate Algebra

Question

Given a quadratic function of the form \(f(x)=a(x-h)^{2}+k,\) answer the following. How do you know if the parabola is narrower than the graph of \(y=x^{2} ?\)

Step-by-Step Solution

Answer

To determine if the given quadratic function \(f(x) = a(x-h)^2 + k\) is narrower than the graph of \(y = x^2\), compare the absolute value of the coefficient \(a\). If the absolute value of \(a\) is greater than 1, the parabola is narrower than the graph of \(y=x^2\); otherwise, it is wider or has the same shape.

Step 1: Identify the key features of the given quadratic function

Recall that the vertex form of a quadratic function is given by \(f(x)=a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola, and \(a\) determines the shape and orientation of the parabola.

Step 2: Determine the effect of the value of a on the parabola's shape

The value of \(a\) influences the shape of the parabola. If \(a>1\), the parabola will be narrower than the graph of the basic quadratic function \(y=x^2\). If \(0

Step 3: Compare the given function to the graph of y=x^2

To know if the parabola is narrower than the graph of \(y = x^2\), we need to compare the value of \(a\) of the given function with \(1\). If the absolute value of \(a\) is greater than 1 (i.e., either \(a > 1\) or \(a < -1\)), the parabola will be narrower than the graph of \(y=x^2\). Otherwise, it will be wider or have the same shape as the graph of \(y = x^2\). In conclusion, to determine if the given quadratic function \(f(x) = a(x-h)^2 + k\) is narrower than the graph of \(y = x^2\), compare the absolute value of the coefficient \(a\). If the absolute value of \(a\) is greater than 1, the parabola is narrower than the graph of \(y=x^2\); otherwise, it is wider or has the same shape.

Key Concepts

Vertex Form

The vertex form of a quadratic function is an essential representation that makes it easy to identify the vertex of a parabola. In mathematical terms, it is expressed as \(f(x) = a(x-h)^2 + k\). Here, the values \(h\) and \(k\) directly give us the coordinates of the vertex of the parabola, \((h, k)\). This form is incredibly useful because it shows at a glance the vertex, which is a crucial point in the graph where the parabola changes direction.

• The vertex \((h, k)\) helps to locate the position of the parabola on the graph.
• The value of \(h\) shifts the parabola along the x-axis.
• The value of \(k\) shifts the parabola along the y-axis.

This form makes it straightforward to determine the vertex, which can simplify both graphing and solving quadratic functions. Understanding this foundation paves the way for further exploration of the parabola's properties.

Parabola Shape

When we talk about the shape of a parabola in a quadratic function, we refer to how "wide" or "narrow" it appears on the graph. This shape is highly influenced by the coefficient \(a\) in the vertex form equation \(f(x) = a(x-h)^2 + k\).

A default parabola, represented by \(y = x^2\), is a symmetrical curve with a specific width. When \(|a| > 1\), the parabola becomes narrower compared to \(y = x^2\). Conversely, if \(0 < |a| < 1\), the parabola is wider.

• Narrower parabola: Occurs when \(|a| > 1\).
• Wider parabola: Happens when \(0 < |a| < 1\).

Additionally, if \(a\) is positive, the parabola opens upwards, resembling a "U" shape. If \(a\) is negative, it opens downwards, like an upside-down "U". This understanding of the shape helps in sketching and interpreting the graphs of quadratic functions effectively.

Coefficient Influence

The coefficient \(a\) in the vertex form of a quadratic function \(f(x) = a(x-h)^2 + k\) greatly influences the parabola's graph. It affects both the "sharpness" of the curve and its orientation.

Here's how the value of \(a\) changes the graph:

• \(a > 1\) or \(a < -1\): The parabola is narrow compared to \(y = x^2\).
• \(-1 < a < 1\): The parabola is wider than \(y = x^2\).
• \(a < 0\): The parabola flips, opening downwards.
• \(a = 1\) or \(a = -1\): The parabola has the same width as \(y = x^2\), either upwards or downwards.

Understanding how \(a\) manipulates the graph lets you predict changes in the width and direction of a parabola. It is a powerful tool when trying to model real-world problems using quadratic equations.

Graph Comparison

Comparing the graph of a given quadratic function in vertex form to the standard parabola \(y = x^2\) helps determine specific characteristics like width and orientation. The key is to examine the absolute value of the coefficient \(a\) in \(f(x) = a(x-h)^2 + k\).

To make this comparison:

• If \(|a| > 1\), the parabola is narrower than \(y = x^2\).
• If \(0 < |a| < 1\), the parabola is wider.
• If \(|a| = 1\), the parabola has the same width as \(y = x^2\).

This comparative approach provides a quick way to identify how a quadratic graph differs in shape and size from the standard reference graph \(y = x^2\). Recognizing these differences is crucial, especially when interpreting and predicting the real-life implications of quadratic models.

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Problem 5

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Problem 6

Other exercises in this chapter

Problem 5

Graph each function by plotting points, and identify the domain and range. $$g(x)=x^{2}-4$$

View solution

Problem 5

a) Does the function attain a minimum or maximum value at its vertex? b) Find the vertex of the graph of \(f(x)\) c) What is the minimum or maximum value of the

View solution

Problem 6

For each pair of functions, find a) \((f g)(x)\) and \(b\) ) \((f g)(-3)\). $$f(x)=-2 x, g(x)=3 x+1$$

View solution

Problem 6

Graph each function by plotting points, and identify the domain and range. $$h(x)=(x-2)^{2}$$

View solution

Key Concepts

Vertex Form

The vertex form of a quadratic function is an essential representation that makes it easy to identify the vertex of a parabola. In mathematical terms, it is expressed as \(f(x) = a(x-h)^2 + k\). Here, the values \(h\) and \(k\) directly give us the coordinates of the vertex of the parabola, \((h, k)\). This form is incredibly useful because it shows at a glance the vertex, which is a crucial point in the graph where the parabola changes direction.

• The vertex \((h, k)\) helps to locate the position of the parabola on the graph.
• The value of \(h\) shifts the parabola along the x-axis.
• The value of \(k\) shifts the parabola along the y-axis.

This form makes it straightforward to determine the vertex, which can simplify both graphing and solving quadratic functions. Understanding this foundation paves the way for further exploration of the parabola's properties.

Parabola Shape

When we talk about the shape of a parabola in a quadratic function, we refer to how "wide" or "narrow" it appears on the graph. This shape is highly influenced by the coefficient \(a\) in the vertex form equation \(f(x) = a(x-h)^2 + k\).

A default parabola, represented by \(y = x^2\), is a symmetrical curve with a specific width. When \(|a| > 1\), the parabola becomes narrower compared to \(y = x^2\). Conversely, if \(0 < |a| < 1\), the parabola is wider.

• Narrower parabola: Occurs when \(|a| > 1\).
• Wider parabola: Happens when \(0 < |a| < 1\).

Additionally, if \(a\) is positive, the parabola opens upwards, resembling a "U" shape. If \(a\) is negative, it opens downwards, like an upside-down "U". This understanding of the shape helps in sketching and interpreting the graphs of quadratic functions effectively.

Coefficient Influence

The coefficient \(a\) in the vertex form of a quadratic function \(f(x) = a(x-h)^2 + k\) greatly influences the parabola's graph. It affects both the "sharpness" of the curve and its orientation.

Here's how the value of \(a\) changes the graph:

• \(a > 1\) or \(a < -1\): The parabola is narrow compared to \(y = x^2\).
• \(-1 < a < 1\): The parabola is wider than \(y = x^2\).
• \(a < 0\): The parabola flips, opening downwards.
• \(a = 1\) or \(a = -1\): The parabola has the same width as \(y = x^2\), either upwards or downwards.

Understanding how \(a\) manipulates the graph lets you predict changes in the width and direction of a parabola. It is a powerful tool when trying to model real-world problems using quadratic equations.

Graph Comparison

Comparing the graph of a given quadratic function in vertex form to the standard parabola \(y = x^2\) helps determine specific characteristics like width and orientation. The key is to examine the absolute value of the coefficient \(a\) in \(f(x) = a(x-h)^2 + k\).

To make this comparison:

• If \(|a| > 1\), the parabola is narrower than \(y = x^2\).
• If \(0 < |a| < 1\), the parabola is wider.
• If \(|a| = 1\), the parabola has the same width as \(y = x^2\).

This comparative approach provides a quick way to identify how a quadratic graph differs in shape and size from the standard reference graph \(y = x^2\). Recognizing these differences is crucial, especially when interpreting and predicting the real-life implications of quadratic models.