Problem 71 - Algebra for College Students

Question

Graph \(y=x^{2}, y=\frac{1}{2} x^{2},\) and \(y=2 x^{2}\) on the same coordinate system. What can you say about the graph of \(y=a x^{2} ?\)

Step-by-Step Solution

Answer

The graph of \(y = a x^{2}\) is a parabola that opens upwards. If \(0 < a < 1\), it is wider than \(y = x^{2}\); if \(a > 1\), it is narrower.

Step 1 - Plot the graph of y=x^{2}

Start by plotting the basic quadratic function, \(y = x^{2}\), on a coordinate system. This function is a parabola that opens upwards with its vertex at the origin (0,0). For example, when \(x = -2\), \(y = 4\); when \(x = -1\), \(y = 1\); when \(x = 0\), \(y = 0\); when \(x = 1\), \(y = 1\); and when \(x = 2\), \(y = 4\).

Step 2 - Plot the graph of y=\frac{1}{2}x^{2}

Next, plot the graph of \(y = \frac{1}{2}x^{2}\). This function is also a parabola that opens upwards but is wider than \(y = x^{2}\). Use the same x-values: when \(x = -2\), \(y = 2\); when \(x = -1\), \(y = 0.5\); when \(x = 0\), \(y = 0\); when \(x = 1\), \(y = 0.5\); and when \(x = 2\), \(y = 2\).

Step 3 - Plot the graph of y=2x^{2}

Now, plot the graph of \(y = 2x^{2}\). This function results in a parabola that opens upwards but is narrower than \(y = x^{2}\). Using the same x-values: when \(x = -2\), \(y = 8\); when \(x = -1\), \(y = 2\); when \(x = 0\), \(y = 0\); when \(x = 1\), \(y = 2\); and when \(x = 2\), \(y = 8\).

Step 4 - Compare the graphs

Compare the three graphs. Notice that the function \(y = a x^{2}\) changes the width of the parabola. If \(0 < a < 1\), the parabola is wider than \(y = x^{2}\). If \(a > 1\), the parabola is narrower. The sign and value of the coefficient 'a' determine how 'stretched' or 'compressed' the graph is.

Key Concepts

Parabolas

Parabolas are U-shaped curves that can open either upward or downward. The standard form of a quadratic equation is written as \(y = ax^2 + bx + c\). In the simplest form, when \(b = 0\) and \(c = 0\), the equation reduces to \(y = ax^2\). The direction in which the parabola opens depends on the coefficient \(a\)… if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
Parabolas have certain distinct features:

• A symmetry axis that passes through its vertex
• A vertex, which is the highest or lowest point
• They are always symmetric about a vertical line through the vertex

When graphing parabolas, it's essential to recognize these features, as they help in sketching and understanding the graph's behavior.

Vertex

The vertex of a parabola is a critical point as it represents the maximum or minimum value of the quadratic function. For a parabolic function in the form \(y = ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\) for the x-coordinate, and then substituting this back into the function to get the y-coordinate. For our simpler equations \(y = ax^2\), the vertex is at the origin (0, 0).
The vertex is particularly useful when comparing different quadratic functions because it remains the same while the shape and width of the parabola change depending on the coefficient \(a\). Understanding the vertex allows for quicker and more accurate graph sketches, aiding in visual comparisons of various parabolic graphs.

Coordinate Systems

Coordinate systems are the framework we use to visualize and graph functions. For parabolas, we typically use a Cartesian coordinate system, which consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is defined by a pair of coordinates \( (x,y) \).
When graphing quadratic functions, choose a set of x-values and compute the corresponding y-values. For the function \(y = x^2\), we used points such as (0, 0), (1, 1), and (2, 4). Plotting these points and connecting them smoothly helps in drawing a parabola. This same process applies to any quadratic function, though the shapes of the parabolas may differ based on the coefficient \(a\). Efficient use of the coordinate system enables accurate and clear visualization of the quadratic functions.

Coefficient Impact on Graph

The coefficient \(a\) in the quadratic function \(y = ax^2\) has a significant impact on the shape of the graph. Comparing three functions, \(y = x^2\), \(y = \frac{1}{2}x^2\), and \(y = 2x^2\), illustrates this effect clearly:

• For \(y = x^2\), the parabola has a standard width.
• For \(y = \frac{1}{2} x^2\), where \(0 < a < 1\), the parabola is wider.
• For \(y = 2x^2\), where \(a > 1\), the parabola is narrower.

The value of \(a\) determines how 'stretched' or 'compressed' the parabola is. Furthermore, if \(a < 0\), the parabola would open downwards. Understanding the impact of the coefficient \(a\) allows for predicting and explaining changes in the graph's shape, enabling more precise graphing and analysis of quadratic functions.