A function f(x) is graphed what is the slope of the function

So, you can build your graph giving values of x say, 1, 2, 3,...and y will always be -4 giving a straight line passing through -4 and parallel to the x axis.

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  Sketching the graph of $f'$

  The graphs of $y=f(x)$ and $y=f'(x)$ are closely related. Since $f'(a)$ represents the slope of the tangent line to $f$ at $x=a$, the slope of $y=f(x)$ at $x$ is the same as the height of the graph $y=f'(x)$ at that $x$.
  For example, if (see the first graph below) the slope of the tangent line to $f$ at $x=7.2$ is $0$, then that means that $f'(7.2)=0$.  What this means on the graph of $f'$ (see the second graph below) is that $f'$ has the height of $0$ at $x=7.2$.  Which is exacly what $f'(7.2)=0$ indicates.  Remember, $f'$ is a function that we have graphed.
  
  
  
  Considering the graphs of $f$ and $f'$ above,
  DO:  Look at the slope of the tangent to $f$ (top graph) at $x=~.8$.  See if the height of $f'$ (bottom graph) is close to the slope you estimated on $f$.  Now approximate the slope of the line tangent to $f$ at $x=3$ (you can use the grid to help approximate this value).  Does it match the height of $f'$ at $x=3$?  Keep playing this game until you are comfortable with the concept.  The easiest points to consider are the places on $f$ where the slope of the tangent is horizontal.  What happens at these $x$-values to $f'$?

  Let f(x) be a function and assume that for each value of x, we can calculate the slope of the tangent to the graph y = f(x)  at x. This slope depends on the value of x that we choose, and so is itself a function. We call this function the derivative of f(x) and denote it by f ´ (x).

  The derivative of f(x) at the point x is equal to the slope of the tangent
  to y = f(x) at x.

  The graph of a function y = f(x) in an interval is increasing (or rising) if all of its tangents have positive slopes. That is, it is increasing if as x increases, y also increases.

  
  
  

  The graph of a function y = f(x) in an interval is decreasing (or falling) if all of its tangents have negative slopes. That is, it is decreasing if as x increases, y decreases.

  
  
  

  The graph of a function y = f(x) has a stationary point at the point where the tangent is horizontal or has zero slope. This always occurs at the points where a function changes from increasing to decreasing and at the points where a function changes from decreasing to increasing. It can also occur at other points and we will discuss this possibility later.

  

  When the graph of a function y = f(x) is vertical or discontinuous the tangent is undefined.

  These basic properties of the derivative are summarized in the following table.

  BehaviourGraphs

  Derivative
  (slope of tangent)

  Graph increasing
  
  f ´(x) > 0
  (positive)Graph decreasing
  
  f ´(x) < 0
  (negative)Tangent horizontal
  
  f ´(x) = 0
  (zero)Tangent vertical
  or No Tangent
  
  f ´(x) undefined

  Example

  Let. Then.
  Sinceover the intervals (-π/2, π/2), (3π/2, 5π/2), and (7π/2, 9π/2), the function is increasing over those intervals.
  Asover the intervals (-3π/2, -π/2), (π/2, 3π/2), and (5π/2, 7π/2) the function is decreasing over those intervals.

  
            

  Exercise 1

  Test that the properties stated in the above table are true. You can examine the fourteen examples provided in the scroll bar on the top of the applet below or enter your own function in the box provided. If you enter your own function, you must use the symbols + for add, − for subtract, * for multiply,  / for divide, and ^ to raise to a power. You can also use various mathematical functions: sin, cos, tan, sec, cot, csc, arcsin, arccos, arctan, exp, ln, log2, log10, abs, sqrt and cubert. (Here, "abs" is the absolute value function, "sqrt" is the square root function and "cubert" is the cube root function.)

  Make sure you understand the following connections between the two graphs.
  • When the graph of the function y= f(x) is horizontal then the graph of its derivative y= f '(x) passes through the x axis (is equal to zero). This occurs only at a stationary point.
  • When the slope of the function y= f(x) is positive, the graph of its derivative y= f '(x) is above the x-axis (is positive).
  • When the slope of the function y= f(x) is negative, the graph of its derivative y= f '(x) is below the x-axis (is negative).
  • When the slope of the function y= f(x) is vertical , the graph of its derivative y= f '(x) is undefined at that value of x.

  Can't see the above java applet? Click here to see how to enable Java on your web browser. (This applet is based on free Java applets from JavaMath )

  
  

  Exercise 2A

  Consider the function f(x) whose graph is shown below.

  
  
  When x < 1, f(x) is increasing decreasingWhen 1< x < 2, f(x) is increasing decreasingWhen x > 2, f(x) is increasing decreasing

  Click on the graph below that is f´(x), the derivative of f(x).

  (a)   (b) (c) (d)

  Example

  Consider the rational function f(x) =. Its graph is a hyperbola with asymptote at x = 4. It is discontinuous at x = 4.

  

  Differentiating, f '(x) =. When x = 4, f '(x) = 1/0, which is not defined. All other values of x have a negative derivative and an associated decreasing slope on the graph. The graph of the derivative is below.

  How do you find a slope from a graph?

  Using the Slope Equation.

  Pick two points on the line and determine their coordinates..

  Determine the difference in y-coordinates of these two points (rise)..

  Determine the difference in x-coordinates for these two points (run)..

  Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope)..

  What is the slope of f x?

  Since f′(a) represents the slope of the tangent line to f at x=a, the slope of y=f(x) at x is the same as the height of the graph y=f′(x) at that x. For example, if (see the first graph below) the slope of the tangent line to f at x=7.2 is 0, then that means that f′(7.2)=0.

  What is the slope y =

  The slope is -1.