Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of the quadratic function with the graph of \(y=x^{2}\). $$f(x)=5 x^{2}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 5x^{2}\) is a narrower and steeper version of the graph of \(y = x^{2}\) since the coefficient of \(x^{2}\) is five times larger, although they both have the same vertex at the origin and open upwards.

Step 1: Graph both functions

The first step when tasked with comparing two functions is to graph the functions. Begin by graphing both functions \(y = x^{2}\) and \(f(x) = 5x^{2}\). The graph of \(y = x^{2}\) is a standard parabola shape which starts at the origin and reaches upwards symmetrically on both sides of the y-axis.

Step 2: Analyzing the shape of \(f(x) = 5x^{2}\)

Next, graph the function \(f(x) = 5x^{2}\). This is a parabola that also opens upward, but it is narrower than the graph of \(y = x^{2}\). This is because the coefficient 5 multiplies the \(x^{2}\), which makes each \(x\) value 5 times larger when squared, and therefore results in a narrower and steeper parabola.

Step 3: Comparing the position

Both parabolas have their vertex at the origin (0,0). So, their position is the same relative to the origin. However, the graph of \(f(x) = 5x^{2}\) rises much faster, due to the 5 multiplying the \(x^{2}\) term, and is therefore narrower than \(y = x^{2}\)

Key Concepts

Graphing Parabolas

When we talk about graphing parabolas, we are referring to the visual representation of quadratic functions. A parabola is a U-shaped curve that can open either upwards or downwards. The basic quadratic function is represented as \(y = x^2\), and its graph is a parabola that opens upwards with its vertex at the origin \((0, 0)\).

To graph a quadratic function, follow these simple steps:

• Identify the vertex. For the basic parabola \(y = x^2\), this is at \((0,0)\).
• Determine the direction of opening. If the \(a\) value (this is the coefficient of \(x^2\)) is positive, the parabola opens upwards. If it is negative, it opens downwards.
• Identify the axis of symmetry, which is a vertical line that passes through the vertex. This line has the equation \(x = h\), where \(h\) is the x-coordinate of the vertex.
• Choose a few x-values to substitute into the equation to find corresponding y-values. Plot these points to form the parabola's shape.

Understanding these basic steps can help you graph any quadratic function successfully.

Vertex Form

Quadratic functions can also be expressed in what is known as the vertex form: \(y = a(x-h)^2 + k\). This form is particularly useful for identifying the crucial features of a parabola.

Here is why:

• Vertex: The vertex of the parabola is easily identified from \((h, k)\) in the equation. This tells us the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
• Vertex Form vs. Standard Form: The vertex form \(y = a(x-h)^2 + k\) contrasts with the standard form \(y = ax^2 + bx + c\). Converting a quadratic function from standard to vertex form involves completing the square.
• Direction and Stretching: The coefficient \(a\) gives us information about the parabola's direction and how stretched or compressed it is. A negative \(a\) value flips the parabola, while a larger or smaller \|a\| value compresses or stretches it.

The vertex form simplifies many tasks, such as determining the parabola's maximum or minimum values and quickly sketching its graph.

Effect of Coefficients on Graph Shape

The coefficients in a quadratic equation significantly affect the graph's shape. Let's take a closer look at how these coefficients influence the parabola.

• Leading Coefficient \(a\): This determines the parabola's width and the direction it opens. For example, in the function \(f(x) = 5x^2\), the \(a\) coefficient is 5, making the parabola narrower than the standard \(y = x^2\) since the parabola is steeper.
• The Vertex: If the equation is in vertex form, the coefficients \(h\) and \(k\) directly affect the vertex location \((h, k)\). This shifts the parabola along the x-axis and y-axis.
• Impact of Changing \(a\): Increasing the absolute value of \(a\) results in a steeper, narrower parabola, while decreasing it leads to a flatter, wider parabola. Negative values of \(a\) indicate the parabola will open downwards.

Understanding these effects helps predict and visualize a parabola's shape before even graphing it.