Question

part a: knowledge & understanding 1. which graph has all four of the following characteristics? - the graph is symmetric with respect to the y - axis. - the graph has a vertical asymptote at ( x = 1 ) - the graph has a horizontal asymptote at ( y = 0 ) - the graph is decreasing on the interval ( (3, infty) ) (there are also some hand - drawn graphs labeled a, b, c, d with yellow markings and some hand - written notes like ( va = 1 ), ( ha = 0 ))

Explanation:

To solve this, we analyze each characteristic and match with the graphs:

Step 1: Analyze "Symmetric about y - axis"

A graph symmetric about the \( y \) - axis satisfies \( f(-x)=f(x) \). This means for every \( x \), the point \( (-x,y) \) is also on the graph.

Step 2: Analyze "Vertical Asymptote at \( x = 1 \)"

A vertical asymptote at \( x = 1 \) means the graph approaches \( \pm\infty \) as \( x \to 1^+ \) or \( x \to 1^- \), so the graph should have a break near \( x = 1 \).

Step 3: Analyze "Horizontal Asymptote \( y = 0 \)"

A horizontal asymptote \( y = 0 \) means as \( x\to\pm\infty \), the graph approaches \( y = 0 \).

Step 4: Analyze "Decreasing on \( (3,\infty) \)"

On the interval \( (3,\infty) \), as \( x \) increases, \( y \) should decrease.

Now, let's check the graphs:

• Symmetry about \( y \) - axis: The graph should be mirror - imaged over the \( y \) - axis.
• Vertical Asymptote at \( x = 1 \): The graph should have a vertical asymptote (a vertical line that the graph approaches) at \( x = 1 \).
• Horizontal Asymptote \( y = 0 \): As \( x \) gets very large (positive or negative), the graph should get close to \( y = 0 \).
• Decreasing on \( (3,\infty) \): For \( x>3 \), as \( x \) increases, \( y \) decreases.

Looking at the options, the graph labeled \( D \) (the one with the circle around it) should satisfy all these conditions. Let's verify each condition for graph \( D \):

• Symmetry about \( y \) - axis: The left and right sides of the \( y \) - axis are mirror images.
• Vertical Asymptote at \( x = 1 \): The graph has a vertical asymptote at \( x = 1 \) (the graph approaches this line but never touches it).
• Horizontal Asymptote \( y = 0 \): As \( x\to\pm\infty \), the graph approaches \( y = 0 \).
• Decreasing on \( (3,\infty) \): For \( x>3 \), as \( x \) increases, the \( y \) - values decrease.

Answer:

The graph with the circle around it (the fourth graph, labeled \( D \)) is the one that has all four characteristics.