Problem 83 - Algebra and Trigonometry

Question

determine whether eamakes sensech statement makes sense or does not make sense, and explain your reasoning. I graphed a hyperbola centered at the origin that had \(y\) -intercepts but no \(x\) -intercepts.

Step-by-Step Solution

Answer

The statement 'I graphed a hyperbola centered at the origin that had \(y\)-intercepts but no \(x\)-intercepts' does make sense because it describes a situation in which a hyperbola is vertically oriented.

Step 1: Understanding Hyperbola Orientation

If a hyperbola is centered at the origin, it will have intercepts either on the \(x\)-axis or \(y\)-axis, but not on both. This is due to its particular shape and orientation, determined by the equation governing the hyperbola. Hyperbolas with a principal axis along the \(y\)-axis will have \(y\)-intercepts, whereas ones aligned along the \(x\)-axis will have \(x\)-intercepts.

Step 2: Analyzing the Statement

Looking at the statement 'I graphed a hyperbola centered at the origin that had \(y\)-intercepts but no \(x\)-intercepts', we can infer that the hyperbola must be vertically oriented. This is because a vertically oriented hyperbola has \(y\)-intercepts (i.e., it crosses the \(y\)-axis) but does not have \(x\)-intercepts (it does not cross the \(x\)-axis).

Step 3: Verifying the Statement

The above analysis shows that the statement is possible, as long as the hyperbola is vertically oriented. Thus, the statement indeed makes sense in the context of a vertical hyperbola.

Key Concepts

Centered at Origin

When we say a hyperbola is centered at the origin, we mean its central point is at \((0,0)\). This acts as a kind of 'pivot' for the hyperbola.
The standard equation for a hyperbola centered at the origin will look like one of these:- For a horizontal hyperbola: \(x^2/a^2 - y^2/b^2 = 1\)- For a vertical hyperbola: \(y^2/a^2 - x^2/b^2 = 1\)
The part of the hyperbola that extends outwards from the origin is dictated by these equations. It's essential to know which axis the hyperbola opens along in order to predict its intercepts.

Y-intercepts

The concept of y-intercepts refers to the points where a curve or line crosses the \(y\)-axis. These occur when the x-value in the equation is zero.
For hyperbolas with a vertical orientation, y-intercepts are achievable and is an indicator of how the curve is shaped. If our hyperbola equation is \(y^2/a^2 - x^2/b^2 = 1\), setting \(x = 0\) lets us calculate the y-intercepts. This would give us:

• \((0, +a)\)
• \((0, -a)\)

Thus, these y-intercepts do indeed exist for vertically oriented hyperbolas.

X-intercepts

X-intercepts are where the curve meets the \(x\)-axis. They are found by setting the y-value to zero in the equation.
For a hyperbola centered at the origin, whether it has x-intercepts depends on its orientation. In a vertical orientation hyperbola, like \(y^2/a^2 - x^2/b^2 = 1\), the equation does not allow for any real values of x when y equals zero. Thus, it leads to no x-intercepts.
This absence of x-intercepts is precisely why our statement 'the hyperbola has y-intercepts but no x-intercepts' holds true for a vertically aligned hyperbola.

Vertical Orientation

A hyperbola's orientation depends on which term in its equation is positive.
In the form \(y^2/a^2 - x^2/b^2 = 1\), the hyperbola is vertically oriented because the \(y^2\) term is positive, meaning it primarily opens along the y-axis.
To visualize, imagine a pair of mirrored curves stretching away from the center along the vertical line.

• They open outward and upward from the origin if \(y^2\) is the positive term, giving a vertically oriented shape.
• This orientation results in y-intercepts to exist, while x-intercepts do not.

Understanding the equation's structure helps in grasping how the hyperbola sits and centers in the coordinate plane.