Question
A coordinate plane going from negative 5 to 5 in increments of 1 labeled every 5 units on the x axis and from negative 1 to 10 in increments of 1 labeled every 5 units on the y axis with points labeled at left parenthesis 0 comma 1 right parenthesis and at left parenthesis 1 comma 0.5 right parenthesis. A curve is drawn through the 2 points that approaches the x axis but does not touch it.
Write the exponential equation shown in the graph.
the two points marked are 0,1 and 1, 0.5
(1 point)
The equation is $$
.
Answers
GPT-4o mini
To find the exponential equation that passes through the points (0, 1) and (1, 0.5), we can use the general form of an exponential function:
\[
y = ab^x
\]
We know that:
1. When \( x = 0 \), \( y = 1 \). This gives us:
\[
1 = ab^0 \implies 1 = a \implies a = 1
\]
2. When \( x = 1 \), \( y = 0.5 \). This gives us:
\[
0.5 = 1 \cdot b^1 \implies b = 0.5
\]
So, substituting \( a \) and \( b \) into the equation, we get:
\[
y = 1 \cdot 0.5^x
\]
Thus, the exponential equation is:
\[
y = 0.5^x
\]
\[
y = ab^x
\]
We know that:
1. When \( x = 0 \), \( y = 1 \). This gives us:
\[
1 = ab^0 \implies 1 = a \implies a = 1
\]
2. When \( x = 1 \), \( y = 0.5 \). This gives us:
\[
0.5 = 1 \cdot b^1 \implies b = 0.5
\]
So, substituting \( a \) and \( b \) into the equation, we get:
\[
y = 1 \cdot 0.5^x
\]
Thus, the exponential equation is:
\[
y = 0.5^x
\]