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Answer:
To find the linear function that describes the statement "Function has a y-intercept of 1 and x-intercept of -4", we can use the intercept form of a linear equation.
1. **Y-intercept**: The y-intercept is where the function crosses the y-axis. It is given as 1. So, the point (0, 1) lies on the function.
2. **X-intercept**: The x-intercept is where the function crosses the x-axis. It is given as -4. So, the point (-4, 0) lies on the function.
In general, a linear function in intercept form can be written as:
\[ y = mx + c \]
where \( c \) is the y-intercept and \( m \) is the slope of the line.
From the given points:
- The y-intercept \( c = 1 \)
- The x-intercept provides another point (-4, 0).
To find the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{-4 - 0} = \frac{-1}{-4} = \frac{1}{4} \]
Now substitute the slope and y-intercept into the equation:
\[ y = \frac{1}{4}x + 1 \]
Therefore, the linear function that describes the given statement "Function has a y-intercept of 1 and x-intercept of -4" is \( y = \frac{1}{4}x + 1 \).
Answer:
[tex]y=\dfrac{1}{4}x+1[/tex]
Step-by-step explanation:
Features of a Linear Function
The equation of a linear function is
y = mx + b,
where m is the slope and b is the y-intercept.
[tex]\dotfill[/tex]
Slope
The slope or rate of change can be calculated by using two points on the function's graph and using the slope formula.
[tex]slope=\dfrac{y_2-y_1}{x_2-x_1}[/tex],
where the subscripts indicate which coordinate point the value originates from.
[tex]\dotfill[/tex]
Intercepts
For y-intercepts, the x value is always 0. This makes sense since all y-intercepts are directly on the y-axis.
Similarly, x-intercepts have y-values of 0.
[tex]\hrulefill[/tex]
Solving the Problem
The problem tells us that the y-intercept is 1, so b = 1.
y = mx + 1.
They also tell us that the x-intercept is -4, meaning
0 = m(-4) + 1.
We need to find m to find the full equation of the function described.
Intercepts are also points on the graph, so we can write the x and y intercepts as (0,1) and (-4,0).
Now, we can plug them into the slope formula to find m!
[tex]slope=\dfrac{1-0}{0-(-4)} =\dfrac{1}{4}[/tex]
So, our equation for this function is
[tex]y=\dfrac{1}{4}x+1[/tex].
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