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Question: How do you find the slope of a tangent to the graph of the function\[f\left( x \right) = 3 - 5x{\tex...

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

Explanation

Solution

The slope of a line is a number that measures its steepness, usually denoted by the lettermm. 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(x)=3−5x..............................(i)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 functionf(x)=3−5xf\left( x \right) = 3 - 5x at(−1,8)\left( { - 1,8} \right).
So for that we need to find the first derivative of the given function.

f(x)=3−5x f′(x)=−5..........................(ii)  f\left( x \right) = 3 - 5x \\\ f'\left( x \right) = - 5..........................\left( {ii} \right) \\\

So now we got the first derivative as−5 - 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(x)=3−5xf\left( x \right) = 3 - 5x would be constant at all points.
Therefore the slope of a tangent to the graph of the given functionf(x)=3−5x at(−1,8)f\left( x \right) = 3 - 5x{\text{ at}}\left( { - 1,8} \right) would be −5 - 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.