Question Graph the function $$ f(x)=\log _{\frac{1}{2}}(x) $$. You can only graph integer points, for example you cannot plot the point $$ \left(-1, \frac{1}{4}\right) $$. There is one point that can be moved above and below the $$ x $$ axis as needed. Do not move either point off of the graph as you will not be able to click on the point again.

Question

UpStudy AI · Accepted Answer

We’re going to graph the logarithmic function

f(x) = log₍½₎(x)

which has a base of ½ (a number between 0 and 1). This tells you a few important things about the graph:

1. The graph is defined only for x > 0.
2. There is a vertical asymptote at x = 0 (as x approaches 0⁺, f(x) → +∞).
3. Because the base is less than 1, the graph is strictly decreasing. As x increases, f(x) becomes more and more negative.
4. The graph crosses the x-axis when f(x) = 0. (Remember: log₍½₎(1) = 0 since any base raised to the power 0 equals 1.)

Since you can only plot points with both coordinates being integers, choose points that neatly work out. Here are some that do:

• At x = 1 → f(1) = log₍½₎(1) = 0 → Plot (1, 0).

• At x = 2 → Find y such that (½)ʸ = 2. (Notice (½)^(–1) = 2.) So, f(2) = –1 → Plot (2, –1).

• At x = 4 → Find y such that (½)ʸ = 4. (Since (½)^(–2) = 4.) So, f(4) = –2 → Plot (4, –2).

• At x = 8 → (½)^(–3) = 8, so f(8) = –3 → Plot (8, –3).

These points are all with integer coordinates. When connecting them smoothly, remember that the curve:

• Continues upward (to very high positive values) as it nears the y-axis (x = 0) because f(x) increases without bound as x → 0⁺.
• Keeps decreasing as x increases, passing through the points above.

Also note the instruction about one point that can be moved above and below the x-axis: This likely refers to a point near the vertical asymptote or one that helps with adjusting the curve. In our case, you must keep the points (such as (1,0) and (2,–1)) on the graph and not move them off the graph.

So, in summary, to graph f(x) = log₍½₎(x) using only integer coordinate points:
 – Plot the points (1, 0), (2, –1), (4, –2), and (8, –3).
 – Draw a smooth, continuously decreasing curve through these points.
 – Draw in the vertical asymptote at x = 0 (a dashed line), noting that as x → 0⁺, f(x) → +∞.

This will give you the correct graph of f(x) = log₍½₎(x).
To graph $$ f(x) = \log_{\frac{1}{2}}(x) $$ using only integer points: 1. **Plot the Points:** - (1, 0) - (2, -1) - (4, -2) - (8, -3) 2. **Draw the Curve:** - Connect these points with a smooth, decreasing curve. 3. **Vertical Asymptote:** - Draw a vertical dashed line at x = 0, indicating the function approaches positive infinity as x approaches 0 from the right. 4. **Constraints:** - Keep all plotted points on the graph. - Do not move any points off the graph. This will accurately represent the function $$ f(x) = \log_{\frac{1}{2}}(x) $$ using only integer coordinate points.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Pre Calculus Questions