# Download Collected works. Vol.4 Continuous geometry and other topics by von Neumann J. PDF By von Neumann J.

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M = − 14 , P = (−2, 1) ............................................................ In exercises 23–28, determine if the lines are parallel, perpendicular, or neither. 23. y = 3(x − 1) + 2 and y = 3(x + 4) − 1 24. y = 2(x − 3) + 1 and y = 4(x − 3) + 1 25. y = −2(x + 1) − 1 and y = 12 (x − 2) + 3 In exercises 1–10, solve the inequality. 1. 3x + 2 < 8 2. 3 − 2x < 7 26. y = 2x − 1 and y = −2x + 2 3. 1 ≤ 2 − 3x < 6 x +2 ≥0 5. x −4 2 7. x + 2x − 3 ≥ 0 4. −2 < 2x − 3 ≤ 5 2x + 1 6. <0 x +2 2 8. x − 5x − 6 < 0 27.

Notice that not every curve is the graph of a function, since for a function, only one y-value can correspond to a given value of x. You can graphically determine whether a curve is the graph of a function by using the vertical line test: if any vertical line intersects the graph in more than one point, the curve is not the graph of a function, since in this case, there are two y-values for a given value of x. 18b correspond to functions. 5 intersects the circle twice. 18b is the graph of a function, even though it swings up and down repeatedly.

2. 30b Cubic: no max or min, a3 < 0 .. Graphing Calculators and Computer Algebra Systems Sketching the Graph of a Cubic Polynomial Sketch a graph of the cubic polynomial f (x) = x 3 − 20x 2 − x + 20. 31b. However, you should recognize that neither of these graphs looks like a cubic; they look more like parabolas. To see the S-shape behavior in the graph, we need to consider a larger range of x-values. To determine how much larger, we need some of the concepts of calculus. For the moment, we use trial and error, until the graph resembles the shape of a cubic.