Question

How do you find the slope of a tangent to the graph of the function\[f\left( x \right) = 3 - 5x{\text{ at}}\left( { - 1,8} \right)\]?

Answer 1

Hint: The slope of a line is a number that measures its steepness, usually denoted by the letter$m$. A Tangent Line is a line which locally touches a curve at one and only one point. The first derivative is an equation for the slope of a tangent line to a curve at an indicated point. Such that to find the slope of a tangent to the graph of the function we need to find the first derivative of the given function.

Complete step by step solution:
Given
\[f\left( x \right) = 3 - 5x..............................\left( i \right)\]
Now we need to find the slope of a tangent to the graph of the given function\[f\left( x \right) = 3 - 5x\] at\[\left( { - 1,8} \right)\].
So for that we need to find the first derivative of the given function.
\[
  f\left( x \right) = 3 - 5x \\
  f'\left( x \right) = - 5..........................\left( {ii} \right) \\
 \]
So now we got the first derivative as$ - 5$.

Since there are no variables present in the equation for the slope of the tangent, we can say that the tangent to the given function \[f\left( x \right) = 3 - 5x\] would be constant at all points.
Therefore the slope of a tangent to the graph of the given function\[f\left( x \right) = 3 - 5x{\text{ at}}\left( { - 1,8} \right)\] would be $ - 5$.

Note: The slope of a line can be positive, negative, zero or undefined.
i) Positive slope: Here, y increases as x increases, so the line slopes upwards to the right. The slope will be a positive number.
ii) Negative slope: Here, y decreases as x increases, so the line slopes downwards to the right. The slope will be a negative number.
iii) Zero slope: Here, y does not change as x increases, so the line is exactly horizontal. The slope of any horizontal line is always zero.
iv) Undefined slope: When the line is exactly vertical, it does not have a defined slope. The two x coordinates are the same, so the difference is zero.