Find the linear function whose contour map (with contour interval m=6 ) is shown in Figure 22. W

Find the linear function whose contour map (with contour...

Key Concepts

Scaling and Relabeling

Scaling and relabeling in the context of contour maps refer to adjusting the contour interval and the numerical labels on the contours. This process does not alter the underlying linearity of the function but rather modifies its representation by changing the unit difference between the contours. It requires a proportional adjustment of the function’s parameters to match the new contour intervals or labels.

Contour Interval

The contour interval is the numerical difference between the function values of adjacent contour lines on a map. This value indicates the rate at which the function changes. For a linear function, changing the contour interval effectively scales the differences between the contours, which directly affects how steep or gradual the change appears.

Contour Map

A contour map is a diagram that uses lines to connect points that have the same function value. In the context of a linear function, these contours appear as straight, parallel lines. This tool is widely used to represent functions of two variables, allowing for an easy visual interpretation of how the function changes over a region.

Linear Function

A linear function in two variables is an expression of the form f(x, y) = ax + by + c, where a, b, and c are constants. This equation represents a plane in three-dimensional space, and its graph shows a constant rate of change. Since the function is linear, its contour lines are parallel and equally spaced if the gradient is constant.

Transcript

00:01 We want to find the linear function whose contour map is given in figure 25 which has a contour interval of 6 then we want to find what the linear function could be if we instead change the contour interval to be 3 so the thing that will be important is we need to pick out some points from this along with the contour curves so the points i pick picked out was c is equal to 0 and the point 0 negative 1, c is equal to 6 and the point 0 0, and c is equal to 12 and the point 3 0.

00:42 Now recall that a linear function in two dimensions would look something kind of like a x plus b y plus c and if we use these points here we can plug them in to help us solve for each of the variables.

01:06 So what i'm going to first do is go ahead and plug in c is equal to 6.

01:11 So we get 6 is equal to a times 0 plus b times 0 plus c.

01:21 And actually maybe the reason why i picked out these just three points is because we have three variables that we need to solve for.

01:30 So we need three equations in a way.

01:33 So these three sets here are three equations that will help us.

01:42 So the a and b go away and we just get 6 is equal to c.

01:47 So that's good for us so far.

01:51 So we can rewrite this first as a x plus b y plus 6.

02:02 Now let's go ahead and use c is equal to 12.

02:06 So we have 12 is equal to and then the point we're going to use for that is 3 -0 so it's going to be a times 3 plus b times 0 plus 6 so we can go ahead and subtract the 6 over so we get 6 is equal to 3a divide each side by 3 so we get 2 is equal to a so now our function becomes f of x y is equal to 2x plus b y plus 6 and for our last one we can go ahead and plug in when c is equal to 0 so we get 0 is equal to so we chose the point 0 negative 1 so 2 0 plus b times minus 1 plus 6 so we subtract the 6 over negative 6 would be equal to negative b, multiply each side by negative 1, and we get 6 is equal to b...

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