91Ó°ÊÓ

When comparing functions, it's essential to understand their defining characteristics. Here, we work with three quadratic functions:

• Function 1: \( f(x) = x^2 \)
• Function 2: \( g(x) = 3x^2 \)
• Function 3: \( h(x) = \frac{1}{4}x^2 \)

These functions differ by the coefficient in front of the \(x^2\) term. The coefficient affects how steep or flat the parabola is.

Examining the Coefficients

In the function \(g(x) = 3x^2\), the coefficient is 3. This makes the function grow very quickly compared to the other functions.

For \(f(x) = x^2\), the coefficient is 1, giving a moderate growth rate.

For \(h(x) = \frac{1}{4} x^2\), the coefficient is \(\frac{1}{4}\), causing the function to grow the slowest.

By observing these coefficients, we can predict that for any positive or negative value of \(x\), \(g(x)\) will be larger than \(f(x)\), and \(f(x)\) will be larger than \(h(x)\). So, the relationships are:

For \(x > 0\), \(g(x) > f(x) > h(x)\)

For \(x < 0\), \(g(x) > f(x) > h(x)\)

Graphing quadratic functions helps visualize their differences. Here are some tips to ensure accurate graphing:

Vertices and Axes
Each quadratic function \( ax^2 + bx + c \) has a vertex and symmetrical axis. For functions like \(f(x) = x^2\), the vertex is at the origin (0,0), and the axis of symmetry is the y-axis. But in our three functions, we don't have any linear or constant terms, so they all share the same vertex at (0,0).

Shape of Parabolas
When we graph

• \(f(x) = x^2\)
• \(g(x) = 3x^2\)
• \(h(x) = \frac{1}{4} x^2\)

on the same grid, their shapes differ due to their coefficients.

The graph of \(h(x) = \frac{1}{4} x^2 \) is the widest and flattest since the coefficient is less than 1.

< li>The graph of \(f(x) = x^2\) is wider but not as flat.

The graph of \(g(x) = 3x^2\) is the narrowest and steepest.

Each parabola opens upwards because all coefficients are positive. By graphing, you get a clear visual understanding of how the coefficients affect the parabolas' shapes and their relative positions.

The coefficient in a quadratic function influences its graph significantly. Here's how:

Impact on Width

A larger coefficient makes the parabola narrower and steeper. For example, \( g(x) = 3x^2 \) rises quickly because the coefficient is 3.

A smaller coefficient results in a wider, flatter parabola. For instance, \( h(x) = \frac{1}{4} x^2 \) doesn't rise as fast due to its smaller coefficient.

When coefficients are compared:

• Coefficients greater than 1 lead to steeper graphs.
• Coefficients between 0 and 1 produce shallower graphs.

This explains why our functions compare as follows:

• For \(x > 0\): \(g(x) > f(x) > h(x)\)
• For \(x < 0\): \(g(x) > f(x) > h(x)\)

Consistent Comparison
Notice the impact of coefficients does not change with the sign of \(x\). Regardless of whether \(x\) is positive or negative, the comparison among the functions remains consistent: \(g(x) > f(x) > h(x)\).

By understanding coefficient effects, graphing quadratics becomes much more predictable.