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Examples • 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 functionsIf the graph of y = f (x) is translated a units horizontally and b units vertically, then the equation of the translated graph is y − b = f(x − a) For in a translation, every point on the graph moves in the same manner Let (x 1, y 1), then, be the coördinates of any point on the graph of yGraphs A quadratic function is one of the form f(x) = ax 2 bx c, where a, b, and c are numbers with a not equal to zero The graph of a quadratic function is a curve called a parabolaParabolas may open upward or downward and vary in width or steepness, but they all

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Y=f(x) graph examples-If we have two variables in a function, one is the dependent variable, another is the independent variable, for example, \(y=f(x)\), we can sketch the graph using two dimensions or axis in the Cartesian plane with coordinates \((x,y)\) All points on the graph of \(y=f(x)\) satisfy the equation \(y=f(x)\) While during the exam writing yourIn our example above, x is the independent variable and y is the dependent variable A function is an equation that has only one answer for y for every x A function assigns exactly one output to each input of a specified type It is common to name a function either f(x) or g(x) instead of y f(2) means that we should find the value of our

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Graph of z = f(x,y) New Resources Quiz Graphing Exponential Functions (Transformations Included)The graph of f(x) in this example is the graph of y = x 2 3 It is easy to generate points on the graph It is easy to generate points on the graph Choose a value for the first coordinate, then evaluate f at that number to find the second coordinateThe graph of y = f (x) is the graph of y = f (x) reflected about the yaxis Here is a picture of the graph of g(x) =(05x)31 It is obtained from the graph of f(x) = 05x31 by reflecting it in the yaxis Summary of Transformations To graph Draw the graph of f and Changes in the equation of y = f(x) Vertical Shifts y = f (x) c

Y = f (x) Process Outcome a Result of Process Inputs The mathematical term Y = f (x), which translates as simply " Y is a function of x," illustrates the idea that the important process outcomes ( Y s) are a result of the drivers ( x 's) within processes The goal of DMAIC is to identify which few process and input variables mainlyThe yintercept formula says that the yintercept of a function y = f(x) is obtained by substituting x = 0 in it Using this, the yintercept of a graph is the point on the graph whose xcoordinate is 0 ie, just look for the point where the graph intersects the yaxis and it is the yinterceptIntegration and the Area Function The area between the graph of the function y = f (x) and the xaxis, starting at x = 0 is called the area function A (x) Example Find the area under the graph y = 2x between x = 2 and x = 4 The area between 2 and 4 can be described as area between x = 0 and x = 4 minus the area between x = 0 and x = 2 y = 2x

Two examples of reflecting a basic graph About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy &However, unlike the graph f(x) = x^4, shown with blue lines, the graph of f(x) = x^3, shown with green lines, has negative ycoordinates ThisStart with a graph of y = f(x) Now replace that with a graph of y = f(x) What you have just done is to "flip the graph upside down", pivoting it on the horizontal xaxis OK, back to square one Start with a graph of y = f(x) Now replace that with a graph of y = f(x)

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For example, the black dots on the graph in the graph below tell us that f (0) = 2 f ( 0) = 2 and f (6) = 1 f ( 6) = 1 However, the set of all points (x,y) ( x, y) satisfying y =f (x) y = f ( x) is a curve The curve shown includes (0,2) ( 0, 2) and (6,1) ( 6, 1) because the curve passes through those pointsExample 1 From the graph of f ( x ), draw a graph of its derivative f ' ( x ) Since f is a line, its slope is constant Moreover Since f is sloping upward, its slope is a positive constant This means f ' ( x) is a positive constant function Show Next StepLet us start with a function, in this case it is f(x) = x 2, but it could be anything f(x) = x 2 Here are some simple things we can do to move or scale it on the graph We can move it up or down by adding a constant to the yvalue g(x) = x 2 C Note to move the line down, we use a negative value for C C >

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All real y ≥ 0 Example a State the domain and range of y = √ x4 b Sketch, showing significant features, the graph of y = √ x41, a>1, a >Y = f (x) = px q It has one independent and one dependent variable The independent variable is x and the dependent one is y P is the constant term or the yintercept and is also the value of the dependent variable When x = 0, q is the coefficient of the

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The graph of #f(x)=x^2# is called a Parabola It looks like this One of the ways to graph this is to use plug in a few xvalues and get an idea of the shape Since the x values keep getting squared, there is an exponential increase on either side of the yaxisGraphs of Functions The coordinate plane can be used for graphing functions To graph a function in the xy plane, we represent each input x and its corresponding output f ( x) as a point ( x, y ), where y = f ( x ) In other words, you use the x axis for the input and the y axis for the output The following video shows how to sketch theOne such example I gave was the following I presented them with the graph $y = f(x)$, where $f(x)$ is the function that takes in the number of miles/hour over the speed limit a driver is going and returns the value of the corresponding fine for speeding in a particular city

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Another fantastic video from wwwMathsMasterOrgExample 1 Graph the logarithmic function f(x) = log 2 x and state range and domain of the function Solution Obviously, a logarithmic function must have the domain and range of (0, infinity) and (−infinity, infinity) Since the function f(x) = log 2 x is greater than 1, we will increase our curve from left to right, a shown below0 moves it up;

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Using Y=f(x) as a concept forces the project manager to consider metrics from the beginning of the project all the way to the end of the project These projects need to show quantitative improvement, so understanding the desired outcome and how inputs affect reaching our project goals is criticalThe range is all real y ≥−3 Example Sketch the graph of f(x)=3x−x2 and find a the domain and range b f(q) c f(x2) –1 0213 1 2 x y –2 x 2 4 6 y 02 4 Mathematics Learning Centre, University of Sydney 4 SolutionThe range of f is given by the interval (∞ , 1 Example 4 Find the domain of function f given below, graph it and find its range f( x ) = √ ( x 2 4) Solution to Example 4 The domain of function given above is found by solving the polynomial inequality x 2 4 ≥ 0 The solution set of the above inequality is given by the interval 2 , 2 which is also the domain of the above function

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The input values will remain the same, so the graph's coordinate points will now be (x, ay) This means that if f (x) = 5x 1 is vertically stretched by a factor of 5, the new function will be equivalent to 5
Again, this graph has the line y = 0 as an asymptote Variations on the General Graph Note that if b is negative, the curve will curve downward as the x values increase Note that if the exponent is negative, the curve will tend upward in the negative x values Consider these basic forms for y = −2 x and y = 2 −x respectivelyEvery point of f has a corresponding image in f' Some useful reflections of y = f(x) are (i) The graph y = −f(x) is the reflection of the graph of f about the xaxis (ii) The graph y = f(−x) is the reflection of the graph of f about the yaxis (iii) The graph of y = f −1 (x) is the reflection of the graph of f in y = x Translation

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Algebra Examples Popular Problems Algebra Graph y=f(x) Find the absolute value vertex In this case, the vertex for is Tap for more steps To find the coordinate of the vertex, set the inside of the absolute value equal to In this case, Replace the variable with in the expressionF (x) Hence, the resulting function is 5 (5x 1) = 25x 5Curves in R2 Graphs vs Level Sets Graphs (y= f(x)) The graph of f R !R is f(x;y) 2R2 jy= f(x)g Example When we say \the curve y= x2, we really mean \The graph of the function f(x) = x2That is, we mean the set f(x;y) 2R2 jy= x2g Level Sets (F(x;y) = c) The level set of F R2!R at height cis f(x;y) 2R2 jF(x;y) = cg Example When we say \the curve x 2 y = 1, we really mean

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Example 1 Graph the point (4, 3) The first number in the parentheses, aka the x value, is 4 This is how far we go horizontally, on the x axis, from (0, 0) Because it's negative, we go left We don't put a point there yet, though We still need to moveSimply put, the Y=f(x) equation calculates the dependent output of a process given different inputs Here's an example of some possible values for the variables in the "breakthrough equation" In a marketing context, the outcome (Y) could equal the number of product sales, where the function (f) could be email opens from a newsletter campaign, and the input (x) could be theSal graphs y=3⋅sin(
⋅x)2 by thinking about the graph of y=sin(x) and analyzing how the graph (including the midline, amplitude, and period) changes as we perform function transformations to get from y=sin(x) to y=3⋅sin(
⋅x)2

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Linear Function Graph has a straight line whose expression or formula is given by;1, the graph strictly increases as xThe most common graphs name the input value x x and the output value y, y, and we say y y is a function of x, x, or y = f (x) y = f (x) when the function is named f f The graph of the function is the set of all points (x, y) (x, y) in the plane that satisfies the equation y = f (x) y = f (x)

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For a function f X → Y the range of f is the set of yvalues such that y = f(x) for some x in X This corresponds to the set of yvalues when we describe a function as a set of ordered pairs (x,y) The function y = √ x has range;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 Example The graph of y = x 1 would be the same as the above graph, but shifted one unit to the right (so the point of the V will hit the xaxis at 1 rather than 0)The graph of any quadratic function f (x) = a x 2 b x c, where a, b, and c are real numbers and a ≠ 0, is called a parabola When graphing a parabola always find the vertex and the yintercept If the xintercepts exist, find those as well

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Therefore, every coördinate pair on the graph of that function is (x, y) = (x, x 2) The ycoördinate is x 2 In other words, the graph of a function y = f(x) is that geometrical figure such that every coördinate pair on it is (x, f(x)) The ycoördinate is f(x) For example, if y = f(x) = x 1, then every coördinate pair on its graph— Graphing \(y=x^2 6 x\) as an example of \(y = a x^2 b x c\) over the domain \(10 \le x \le 10\text{,}\) but with the ability to easily change the domain of the graph Often, when we graph, we will want to change the domain of the graph Most easily, I may want to zoom in on a particular region to get a better view of some interestingSafety How works Test new features ©

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0 moves it downExample 6 Graph f (x) = 1 2 x 1 and g (x) = 3 on the same set of axes and determine where f (x) = g (x) Solution Here f is a linear function with slope 1 2 and yintercept (0,1) The function g is a constant function and represents a horizontal line Graph both ofBasic Examples The graph of an exponential function is a strictly increasing or decreasing curve that has a horizontal asymptote Let's find out what the graph of the basic exponential function y = a x y=a^x y = ax looks like (i) When a >

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