Piecewise

f(x) = { (x^2, x<=2) (mx + b, x>2)

I am not sure what the graph would look like for the 2nd function.

Any tips on drawing functions like this would be of great help.

There is no unique graph for y = mx + b without knowing the values of m and b .

The only thing I can surmise is that the straight line will continue on from (2,4), which is the end point of the parabola.

so the endpoint of y = x^2 , x ≤ 2 is (2,4)
dy/dx of the parabola is 2x
at (2,4), dx/dy = 4

so find y = mx + b, with m = 4 and point (2,4)
4 = 4(2) + b
b = -4

so graph y = x^2 up to x = 2, then from there
graph y = 4x - 4 , but only for x > 2

(You can use the y-intercept of -4 and the point (2,4) to get the straight line, but don't draw it for x < 2 )

To understand the graph of the second function in the given piecewise function, f(x) = { (x^2, x<=2), (mx + b, x>2), we need to consider the two cases separately.

Case 1: x <= 2
For x values less than or equal to 2, the function is defined as f(x) = x^2. This is a standard quadratic function. The graph of f(x) = x^2 is a parabola that opens upwards. It has its vertex at the origin (0,0), and as x increases, the function value increases. So, for x values less than or equal to 2, the graph of the function will be a parabolic curve.

Case 2: x > 2
For x values greater than 2, the function is defined as f(x) = mx + b. This is a linear function (a straight line). To draw the graph of this linear segment, we need to know the values of the slope, m, and y-intercept, b.

Once you have the values of m and b, follow these steps to draw the graph:

1. Plot the y-intercept: Plot the point (0, b). This represents the y-intercept, where the line intersects the y-axis.

2. Use the slope to find other points: The slope, m, represents how the line rises or falls as x increases. If the slope is positive, the line will slope upwards to the right; if negative, it will slope downwards to the right. Move horizontally from the y-intercept, and vertically based on the slope, to find at least one more point on the line.

3. Connect the points: Once you have two or more points, draw a straight line passing through them. Make sure the line extends beyond the x > 2 region.

4. Determine the continuity: To determine if the two segments are continuous at x = 2, check whether the function values coincide at that point. Specifically, evaluate x^2 at x = 2 and compare it to the value mx + b at x = 2. If they are equal, the function is continuous at x = 2. If not, there will be a "jump" or discontinuity at x = 2.

With these steps, you should be able to sketch the graph of the piecewise function f(x) = { (x^2, x<=2), (mx + b, x>2), including both the parabolic curve and the linear segment.