Problem 55 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. $$ x=2 y^{2} $$

Step-by-Step Solution

Answer

The graph of \( x = 2y^2 \) is a sideways parabola that opens to the right.

Step 1: Understand the Equation Form

The given equation is \( x = 2y^2 \). This is a quadratic equation in "y". Typically, quadratics are expressed as \( y = ax^2 + bx + c \), but here it's reversed, meaning for each value of \( y \), there is a corresponding \( x \) value. This means the graph will be a sideways parabola.

Step 2: Identify the Graphing Format

The equation \( x = 2y^2 \) can be rearranged to express \( y \) in terms of \( x \). However, for graphing, it's often useful to test some values of \( y \) and determine corresponding \( x \). This highlights the curve symmetry about the x-axis due to \( y^2 \) which always yields a non-negative value.

Step 3: Choose the Range and Calculate Points

Select values for \( y \). Typically, a range from \(-3\) to \(3\) is considered. For these, compute: \( y = -3, x = 2(-3)^2 = 18 \); \( y = -2, x = 2(-2)^2 = 8 \); \( y = -1, x = 2(-1)^2 = 2 \); \( y = 0, x = 2(0)^2 = 0 \); and mirror values for positive \( y \). This gives points like \((18,-3), (8,-2), (2,-1), (0,0), (2,1), (8,2), (18,3) \).

Step 4: Plotting the Points

Using a graphing calculator, manually plot these calculated points \((18,-3), (8,-2), (2,-1), (0,0), (2,1), (8,2), (18,3)\). These points form the shape of a parabola opening horizontally (left to right) along the \( x \)-axis.

Step 5: Draw the Parabola

Connect the plotted points smoothly to graph a sideways parabola. It should appear as U-shaped and symmetric about the \( y \)-axis, indicating \( x = 2y^2 \) forms a parabola that opens to the right.

Key Concepts

Sideways Parabola

When we encounter a quadratic equation like \( x = 2y^2 \), it may seem a bit unusual if you're used to equations like \( y = ax^2 + bx + c \). Here, the equation is reversed, implying for each value of \( y \), we have a corresponding \( x \). Thus, instead of a typical vertical parabola, the graph creates a sideways parabola that opens horizontally.

The sideways opening is due to how \( x \) is expressed as a function of \( y^2 \). The positive coefficient means the parabola will open to the right. As \( y \) becomes positive or negative, \( x \) increases as long as \( y^2 \) yields a non-negative value. This is why the graph of \( x = 2y^2 \) stretches sideways, forming a 'U' shape symmetric about the \( y \)-axis.

Graphing Calculator

A graphing calculator can be an invaluable tool when dealing with equations like this. It allows you to input the equation \( x = 2y^2 \) directly and see the graph take shape almost instantly.

To use a graphing calculator:

• Input the equation into the calculator's graphing function.
• Set an appropriate viewing window that includes the range of points you expect to see, typically from the minimum to the maximum calculated values of \( y \).
• Use the graphing software to plot points manually or view the automatically generated graph for immediate insight into the shape of the parabola.

By visualizing the equation on a calculator, you improve your understanding of how various values of \( y \) relate to \( x \) and how the parabola forms and behaves.

Plotting Points

After understanding the form of your equation, plotting points is the next practical step. It's a methodical way to ensure the accuracy of your graph. Begin by selecting values for \( y \) to calculate corresponding \( x \) values with the equation \( x = 2y^2 \).

Here's a step-by-step guide on plotting these points:

• Select a set of \( y \) values, such as from \(-3\) to \(3\), as this gives a symmetrical view.
• Calculate \( x \) for each \( y \). For instance: \( y = 3, x = 2(3)^2 = 18 \).
• Once you have your coordinates, such as \((18, -3), (8, -2), (2, -1), (0, 0), (2, 1), (8, 2), (18, 3)\), plot them on your graph.

Always remember, these points should reflect the horizontal parabola and help visualize the curve's direction and extent.

Symmetry in Graphs

Symmetry is a crucial concept in graphing, especially when it comes to parabolas, whether they're vertical or horizontal. For the equation \( x = 2y^2 \), symmetry appears about the \( y \)-axis.

Here’s what symmetry involves in this context:

• The parabola is symmetric about the \( y \)-axis because \( y^2 \) results in the same \( x \) value for both positive and negative values of \( y \).
• This means if you fold the graph along the \( y \)-axis, both sides will match, indicating balanced shapes for the generated parabola.
• Because we calculated both positive and negative \( y \) values to find corresponding \( x \), you’ll see that points are mirrored across the \( y \)-axis.

Understanding graph symmetry not only makes plotting easier but also provides insights into the nature of quadratic equations and their solutions.