y + 3 = -1/2 (x - 10) Use your Desmos graphing calculator!

Question

Match "True" to statements that are True and "False" to statements that are not True.

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This is a Linear Function
This graph goes through the point (10, -3)
The Range (y-vales) for this graph is over the interval (- infinity, + infinity)
As x increases, y decreases
The end behavior of this graph is
As x --> - infinity, y --> + infinity and x --> + infinity, y --> - infinity

True False

Answer 1

To address the statements about the linear equation \( y + 3 = -\frac{1}{2} (x - 10) \), we can first rewrite this equation in slope-intercept form (y = mx + b) for clarity.

1. Start with the original equation: \[ y + 3 = -\frac{1}{2} (x - 10) \] Distributing the \(-\frac{1}{2}\): \[ y + 3 = -\frac{1}{2}x + 5 \] Isolating \(y\): \[ y = -\frac{1}{2}x + 5 - 3 \] Simplifying: \[ y = -\frac{1}{2}x + 2 \]

Now we can evaluate each statement based on this equation:

1. This is a Linear Function: True
   The equation is of the form \(y = mx + b\), indicating it is linear.

2. This graph goes through the point (10, -3): False
   To check, substitute \(x = 10\): \[ y = -\frac{1}{2}(10) + 2 = -5 + 2 = -3 \] So, (10, -3) is a point on the line, this part is True.

3. The Range (y-values) for this graph is over the interval (-infinity, +infinity): True
   Linear functions have a range of all real numbers.

4. As x increases, y decreases: True
   The slope is \(-\frac{1}{2}\), which means y decreases as x increases.

5. The end behavior of this graph is: As x → -infinity, y → +infinity and x → +infinity, y → -infinity: True
   The negative slope indicates that as x increases, y decreases, following the described end behavior.

So the answers are:

• This is a Linear Function: True
• This graph goes through the point (10, -3): True
• The Range (y-values) for this graph is over the interval (-infinity, +infinity): True
• As x increases, y decreases: True
• The end behavior of this graph is: As x → -infinity, y → +infinity and x → +infinity, y → -infinity: True

In summary, you would select "True" for all statements as they all are true for this linear function.