If y = f (x), the graph of y = af (x) is ), parallel to the xaxis Scale factor 1/a means that the "stretch" actually causes the graph to be squashed if a is a number greater than 1• The graph of f(x)=x2 is a graph that we know how to draw It's drawn on page 59 We can use this graph that we know and the chart above to draw f(x)2, f(x) 2, 2f(x), 1 2f(x), and f(x) Or to write the previous five functions without the name of the function f,Graph of f (½ x) = (½ x) 2 (This graph grows only half as fast as does the graph of the regular function, shown in the previous box) As you can see, multiplying inside the function (inside the argument of the function) causes the graph to get thinner or fatter
Aim 36 How Do We Determine The Properties Of A Function
What is f on a graph
What is f on a graph-Learning Objectives 451 Explain how the sign of the first derivative affects the shape of a function's graph;E) If f'(x)=0, then the x value is a point of inflection for f To illustrate these principles, consider the following problems 1) Suppose a) On what interval is f increasing?
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Example 1 Fig 1 is the graph of the parabola f ( x) = x2 − 2 x − 3 = ( x 1) ( x − 3) The roots −1, 3 are the x intercepts Fig 2 is its reflection about the xaxis Every point that was above the x axis gets reflected to below the x axis And every point below the x axis gets reflected above the xDefining the Graph of a Function The graph of a function f is the set of all points in the plane of the form (x, f(x)) the graph of f to be the graph of the equation y = f(x) So, the graph of a function if a special case of the graphLinear functions have the form f(x) = ax b, where a and b are constants In Figure 111, we see examples of linear functions when a is positive, negative, and zero Note that if a > 0, the graph of the line rises as x increases In other words, f(x) = ax b is increasing on ( − ∞, ∞)
When x > 0, f(x) is equal to 2x When x = 0, f(x) is equal to 1 When x < 0, f(x) is equal to 2x When given a piecewise function graph, make sure to observe the given intervals where f(x) has varied graphs But before we try out examples that involve analyzing piecewise function graphs, let's go ahead and learn how we can evaluate and graph311 Recognize the meaning of the tangent to a curve at a point 312 Calculate the slope of a tangent line 313 Identify the derivative as the limit of a difference quotient 314 Calculate the derivative of a given function at a point The graphs of f (x) = 1 x f (x) = 1 x and y =Function Transformations Just like Transformations in Geometry, we can move and resize the graphs of functions Let us start with a function, in this case it is f
2) Sketch the graph of a function whose first and second derivatives are always X the inputs, factors or whatever is necessary to get the outcome (there can be more than one possible x) F the function or process that will take the inputs and make them into the desired outcome Simply put, the Y=f(x) equation calculates the dependent output of a process given different inputsFrom the graph of f ' (x), draw a graph of f(x) f ' is negative, then zero, then positive This means f will be decreasing for a bit, and will then turn around and increase We just don't know exactly where the graph of f(x) will be in relation to the yaxisWe can figure out the general shape of f, but f we could take the graph of f that we just made and shift it up or down along the y
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1 Below Is Shown The Graph Of F X
The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x It is called the derivative of f with respect to x If x and y are real numbers, and if the graph of f is plotted against x, derivative is the slope of this graph at each point Begin by evaluating for some values of the independent variable x Figure 251 Now plot the points and compare the graphs of the functions g and h to the basic graph of f(x) = x2, which is shown using a dashed grey curve below Figure 252 The function g shifts the basic graph down 3 units and the function h shifts the basic graph up 3 unitsIt means f(x) is always positive irrespective of the values of x Its graph is of v shape having sharp edge at origin
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A math reflection flips a graph over the yaxis, and is of the form y = f (x) Other important transformations include vertical shifts, horizontal shifts and horizontal compression Let's talk about reflections Now recall how to reflect the graph y=f of x across the x axisUndefined derivatives Note From here on, whenever we say "the slope of the graph of f at x," we mean "the slope of the line tangent to the graph of f at x" In some cases, the derivative of a function f may fail to exist at certain points on the domain of f, or even not at allThat means at certain points, the slope of the graph of f is not welldefinedHence, the name "piecewise" function When I evaluate it at various x values, I have to be careful to plug the argument into the correct piece of the function
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The graph always lies above the xaxis, but becomes arbitrarily close to it for large negative x;453 Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function's graph;It remains to confirm f(x) and g(x) each share exactly one point in common with h(x) Again the tradition is to do so algebraically, but it might be instructive to look at some graphs of h(x) f(x) and h(x) g(x), such as the following Can we immediately generate graphs of other f(x), g(x), and h(x) satisfying the conditions of the problem?
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Algebra Graph f (x)=1 f (x) = 1 f (x) = 1 Rewrite the function as an equation454 Explain the concavity test for a function over an open intervalThis is the graph of y = f(x) First I want to label the coordinates of some points on the graph Since, for each point on the graph, the x and y coordinates are related by y = f(x), I can put the coordinates of these points in a list
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