# Minimum Norm Extremals in Function Spaces by S.W. Fisher, J.W. Jerome By S.W. Fisher, J.W. Jerome

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It is not necessary to remember which sign moves the function which way. The placement of the function on the x-axis comes out of the analysis. x Ϫ(p/2) Ϫ(p/4) y sin (2x ϩ p) 0 1 y = sin( 2 x + p ) p 2 x p _ p _p Figure 1-30 shows a sine function with ampli4 4 2 tude 1. The 2 affects the frequency and the moves the function right or left. Set up the p Fig. 1-30 chart again forcing the argument to be zero or p/2 and determining the appropriate x value. Set 2x ϩ p ϭ 0 and solve for x ϭ Ϫ(p/2). Set 2x ϩ p ϭ p/2 and solve for x ϭ Ϫ(p/4).

5x2 ϩ 1. 5 coefﬁcient), it opens up (positive coefﬁcient of the x2 term), and it is moved up the axis one unit. Now put in some numbers: x ϭ 0, y ϭ 1 is the apex, and the y-axis is the symmetry line. Add the points x ϭ Ϯ2, y ϭ 3 and sketch the graph (Fig. 1-10). Example 1-18 Graph y ϭ Ϫ2x2 Ϫ 2. 5 x 2 + 1 x y = −2 x 2 − 2 Fig. 1-10 Solution: Look at the function and verify the following statement. This is a parabola that opens down, is sharper than normal, and is displaced two units in 13 Mathematical Background the negative direction.

Solve by quadratic formula 2x2 ϩ 5x Ϫ 6 ϭ 0. Write (a ϩ b)3 using the binomial expansion. Convert p/4 rad to degrees. 45 rad to degrees. 6 rad to degrees. Convert 80Њ to radians. Convert 200Њ to radians. Switch the point (Ϫ4,6) to polar form. Switch the point (2,Ϫ5) to polar form. Switch 3 @ 20Њ to rectangular form. Switch 5 @ 60Њ to rectangular form. Solve 6 ϭ ln3x for x. 3. 8. 48 in exponential form. 56. 1)3. 3]. What is the y intercept for the function y ϭ 3x Ϫ 2? What is the y intercept for the function y ϭ x2 ϩ 3?