Problem 6 - 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 wider than the graph of \(y=x^{2} ?\)

Step-by-Step Solution

Answer

If the absolute value of the coefficient "a" for the given function is less than 1 (i.e., \(|a| < 1\)), then the parabola is wider than the graph of \(y = x^2\).

Step 1: Identify the given quadratic function and the standard quadratic function

The given quadratic function is in the form \(f(x)=a(x-h)^2+k\), and the standard quadratic function we'll compare to is \(y=x^2\).

Step 2: Determine the coefficient "a" for both functions

For the given quadratic function, the value of "a" is already provided in the equation, which is just \(a\). For the standard quadratic function, \(y=x^2\), the coefficient for the x² term is 1, which means that \(a=1\).

Step 3: Compare the absolute values of "a" for both functions

To determine if the given parabola is wider than the graph of \(y=x^2\), we need to compare the absolute values of "a" for both functions. Since the standard quadratic function has an "a" value of 1, if the absolute value of "a" of the given function is less than 1, then the parabola will be wider. So, if \( |a| < 1\) for the given function, then it is wider than the graph of \(y=x^2\).

Key Concepts

Parabola Width

When talking about the width of a parabola, we're referring to how "wide" or "narrow" it appears. The parabola's width is intimately linked to the coefficient "a" in the vertex form of a quadratic function. For the standard quadratic function, given as \(y=x^2\), the parabola is a benchmark we often use for comparison. Now, what makes a parabola wider when compared to this standard? It's all in the magic number of the coefficient "a".

• If the absolute value of "a" is less than 1 \((|a|<1)\), the parabola becomes wider.
• Conversely, if \(|a|>1\), the parabola becomes narrower.

Understanding this concept helps you visualize how changes in "a" affect the shape of the parabola. It’s fascinating how such a small number can alter the geometry of a curve in such significant ways!

Coefficient Comparison

In quadratic functions, comparing coefficients is a fundamental task when analyzing and predicting the behavior of parabolas. The coefficient "a" in the vertex form \(f(x)=a(x-h)^2+k\) tells us a lot about the parabola's orientation and width.

• The sign of "a" (positive or negative) indicates if the parabola opens upwards or downwards.
• Comparison with the usual quadratic, \(y=x^2\), means looking at their "a" values.

In the standard form, \(a=1\). By comparing this with our quadratic function's "a", we get insights into how our parabola differs:

• If our parabola's \(|a|<1\), it is wider than the standard.
• If \(|a|>1\), it is narrower, portraying how closely spaced the arms of the parabola are.

Being attentive to these comparisons allows for deeper understanding and greater control when graphing or analyzing quadratic functions.

Vertex Form

Vertex form of a quadratic function \(f(x)=a(x-h)^2+k\) provides valuable insight into the graph's features. Identified as the "vertex form" due to its ability to readily showcase the vertex of the parabola, it's often preferred for graphing and theoretical purposes. Here’s why:

• The vertex of the parabola is given directly by the point \((h, k)\), making it incredibly easy to identify.
• This format highlights how shifts along the x-axis \((h)\) and y-axis \((k)\) alter its position.
• The coefficient "a" guides us on the width and direction (upward or downward) based on its value and sign.

The vertex form is integral for those who want to delve deeper into the nature of quadratic functions, aiding in quick sketching and interpretation of graphs without much algebraic trouble.