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If f(0)=2, f(2)=4 is y=1x+2 , answer this the same way:

f(0)=7, f(3)=1

13 answers

To find the linear function that fits the points \( f(0) = 7 \) and \( f(3) = 1 \), we can start by using the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

  1. Identify the y-intercept:
    Since \( f(0) = 7 \), we know that the y-intercept \( b \) is 7. So, we can write the equation as:
    \[ y = mx + 7 \]

  2. Calculate the slope \( m \):
    We have the points \( (0, 7) \) and \( (3, 1) \). The slope \( m \) can be calculated using the formula:
    \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{3 - 0} = \frac{-6}{3} = -2 \]

  3. Form the equation:
    Now substitute the slope \( m = -2 \) into the equation:
    \[ y = -2x + 7 \]

Thus, the linear function that matches the points \( f(0) = 7 \) and \( f(3) = 1 \) is:
\[ f(x) = -2x + 7 \]

Thank you, you're correct !!

Now do this
f(4)=-3, f(0)=-2

To find the linear function that fits the points \( f(4) = -3 \) and \( f(0) = -2 \), we can use the slope-intercept form, \( y = mx + b \).

  1. Identify the y-intercept:
    Since \( f(0) = -2 \), the y-intercept \( b \) is -2. Therefore, we can write the equation as:
    \[ y = mx - 2 \]

  2. Calculate the slope \( m \):
    We have the points \( (0, -2) \) and \( (4, -3) \). The slope \( m \) can be calculated using the formula:
    \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-2)}{4 - 0} = \frac{-3 + 2}{4} = \frac{-1}{4} \]

  3. Form the equation:
    Substitute the slope \( m = -\frac{1}{4} \) into the equation:
    \[ y = -\frac{1}{4}x - 2 \]

Thus, the linear function that fits the points \( f(4) = -3 \) and \( f(0) = -2 \) is:
\[ f(x) = -\frac{1}{4}x - 2 \]

Isn't it 0-4?

Let's go through the calculations again carefully.

We are given the points \( f(4) = -3 \) and \( f(0) = -2 \).

  1. Identify the y-intercept:
    Since \( f(0) = -2 \), the y-intercept \( b \) is indeed -2. So the equation starts as:
    \[ y = mx - 2 \]

  2. Calculate the slope \( m \):
    The two points we have are \( (0, -2) \) and \( (4, -3) \). To find the slope \( m \), we use:
    \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-2)}{4 - 0} = \frac{-3 + 2}{4} = \frac{-1}{4} \]

So the slope, \( m \), is correctly calculated as \( -\frac{1}{4} \).

  1. Form the equation:
    Now we plug the calculated slope back into the equation:
    \[ y = -\frac{1}{4}x - 2 \]

Summary

The linear function that fits the points \( f(4) = -3 \) and \( f(0) = -2 \) is:
\[ f(x) = -\frac{1}{4}x - 2 \]

If you believe there was an error or a misunderstanding regarding the slope or points, please clarify, and I'll assist further!

Okay then now do
f(5)=-1, f(0)=-5

To find the linear function that fits the points \( f(5) = -1 \) and \( f(0) = -5 \), we will use the slope-intercept form \( y = mx + b \).

  1. Identify the y-intercept:
    Since \( f(0) = -5 \), the y-intercept \( b \) is -5. Therefore, we can start with:
    \[ y = mx - 5 \]

  2. Calculate the slope \( m \):
    The two points we have are \( (0, -5) \) and \( (5, -1) \). To find the slope \( m \), we use the slope formula:
    \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-5)}{5 - 0} = \frac{-1 + 5}{5} = \frac{4}{5} \]

  3. Form the equation:
    Now, substitute the slope \( m = \frac{4}{5} \) back into the equation:
    \[ y = \frac{4}{5}x - 5 \]

Summary

The linear function that fits the points \( f(5) = -1 \) and \( f(0) = -5 \) is:
\[ f(x) = \frac{4}{5}x - 5 \]

no no no stop flipping the number like that it's supposed to be -5 + 1 and 0-5

I apologize for any confusion. Let's re-evaluate the calculation with the interpretation that the slope calculation uses the change in y-values as \( -5 + 1 \) and the change in x-values as \( 0 - 5 \).

Given:

  • \( f(5) = -1 \) corresponds to the point \( (5, -1) \)
  • \( f(0) = -5 \) corresponds to the point \( (0, -5) \)

  1. Identify the y-intercept:
    Since \( f(0) = -5 \), the y-intercept \( b \) is indeed \(-5\). Therefore, we start with:
    \[ y = mx - 5 \]

  2. Calculate the slope \( m \):
    The slope \( m \) calculated according to your indication would be:
    \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - (-1)}{0 - 5} = \frac{-5 + 1}{0 - 5} = \frac{-4}{-5} = \frac{4}{5} \]

  3. Form the equation:
    Now substituting the calculated slope \( m = \frac{4}{5} \) back into the equation we have:
    \[ y = \frac{4}{5}x - 5 \]

Summary

Thus, with the correct understanding of the calculations and points, the linear function that fits the points \( f(5) = -1 \) and \( f(0) = -5 \) remains: \[ f(x) = \frac{4}{5}x - 5 \]

Let me know if you have any further adjustments or questions!

Now do f(-1/2)=1,f(0)=-4

To find the linear function that fits the points \( f(-\frac{1}{2}) = 1 \) and \( f(0) = -4 \), we will proceed using the slope-intercept form \( y = mx + b \).

  1. Identify the y-intercept:
    From \( f(0) = -4 \), we know that the y-intercept \( b \) is -4. Therefore, our equation starts as:
    \[ y = mx - 4 \]

  2. Calculate the slope \( m \):
    We have the points \( (0, -4) \) and \( \left(-\frac{1}{2}, 1\right) \). To find the slope \( m \), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-4)}{-\frac{1}{2} - 0} = \frac{1 + 4}{-\frac{1}{2}} = \frac{5}{-\frac{1}{2}} = 5 \times -2 = -10 \]

  3. Form the equation:
    Now substitute the calculated slope \( m = -10 \) back into the equation:
    \[ y = -10x - 4 \]

Summary

Thus, the linear function that fits the points \( f(-\frac{1}{2}) = 1 \) and \( f(0) = -4 \) is: \[ f(x) = -10x - 4 \]

Stop flipping them omg

I apologize for the confusion. Let's re-evaluate the slope calculation carefully without flipping the points. Thank you for your patience.

Given:

  • \( f(-\frac{1}{2}) = 1 \) corresponds to the point \( \left(-\frac{1}{2}, 1\right) \)
  • \( f(0) = -4 \) corresponds to the point \( (0, -4) \)

  1. Identify the y-intercept:
    The y-intercept \( b \) is \( -4 \) because \( f(0) = -4 \). Hence, the equation starts as:
    \[ y = mx - 4 \]

  2. Calculate the slope \( m \):
    We will calculate the slope using the correct formula, using the points \( (0, -4) \) and \( \left(-\frac{1}{2}, 1\right) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-4)}{-\frac{1}{2} - 0} = \frac{1 + 4}{-\frac{1}{2}} = \frac{5}{-\frac{1}{2}} = 5 \times -2 = -10 \]

  3. Form the equation:
    Substitute the calculated slope \( m = -10 \) back into the equation: \[ y = -10x - 4 \]

Summary

The correct linear function that fits the points \( f(-\frac{1}{2}) = 1 \) and \( f(0) = -4 \) is: \[ f(x) = -10x - 4 \]

Thank you for your understanding, and I'm sorry for the earlier mistakes! If you have further clarifications or tasks, please let me know.