Answer :
To find the equation of a linear function that satisfies the given conditions, [tex]\( f(2) = -1 \)[/tex] and [tex]\( f(0) = 5 \)[/tex], we can follow these steps:
1. Identify the Function Form:
A linear function can be expressed in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
2. Use the Given Information:
We have two pieces of information:
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -1 \)[/tex] (i.e., the point (2, -1)).
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 5 \)[/tex] (i.e., the point (0, 5)).
3. Find the Y-intercept ([tex]\( b \)[/tex]):
Since [tex]\( f(0) = 5 \)[/tex], this tells us directly that the y-intercept, [tex]\( b \)[/tex], is 5.
4. Calculate the Slope ([tex]\( m \)[/tex]):
We can use the slope formula:
[tex]\[
m = \frac{{f(x_2) - f(x_1)}}{{x_2 - x_1}}
\][/tex]
Let's use the points (0, 5) and (2, -1):
[tex]\[
m = \frac{{-1 - 5}}{{2 - 0}} = \frac{{-6}}{2} = -3
\][/tex]
5. Write the Equation:
Now that we have the slope [tex]\( m = -3 \)[/tex] and the y-intercept [tex]\( b = 5 \)[/tex], we can write the equation of the linear function:
[tex]\[
f(x) = -3x + 5
\][/tex]
Therefore, the equation for the linear function that satisfies the given conditions is [tex]\( f(x) = -3x + 5 \)[/tex].
1. Identify the Function Form:
A linear function can be expressed in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
2. Use the Given Information:
We have two pieces of information:
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -1 \)[/tex] (i.e., the point (2, -1)).
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 5 \)[/tex] (i.e., the point (0, 5)).
3. Find the Y-intercept ([tex]\( b \)[/tex]):
Since [tex]\( f(0) = 5 \)[/tex], this tells us directly that the y-intercept, [tex]\( b \)[/tex], is 5.
4. Calculate the Slope ([tex]\( m \)[/tex]):
We can use the slope formula:
[tex]\[
m = \frac{{f(x_2) - f(x_1)}}{{x_2 - x_1}}
\][/tex]
Let's use the points (0, 5) and (2, -1):
[tex]\[
m = \frac{{-1 - 5}}{{2 - 0}} = \frac{{-6}}{2} = -3
\][/tex]
5. Write the Equation:
Now that we have the slope [tex]\( m = -3 \)[/tex] and the y-intercept [tex]\( b = 5 \)[/tex], we can write the equation of the linear function:
[tex]\[
f(x) = -3x + 5
\][/tex]
Therefore, the equation for the linear function that satisfies the given conditions is [tex]\( f(x) = -3x + 5 \)[/tex].
