Question

1. which corner point on this graph will result in the maximum value of the function $p = 3x - y$? a coordinate plane with two horizontal lines. one through (0, -4) and one through (0, 8). a vertical line through (14, 0). a line through ordered pairs (2, 8) and (8, -4). intersection of the lines is marked and the trapezoidal area formed is shaded. options: (8, -4), (14, -4), (2, 8), (14, 8)

Explanation:

Step1: Identify corner points

From the image description, the corner points of the shaded trapezoid are \((2, 8)\), \((14, 8)\), \((14, -4)\), and \((8, -4)\). Wait, actually, let's check the description again: two horizontal lines (one at \(y = 8\), one at \(y=-4\)), vertical line at \(x = 14\), and a line through \((2, 8)\) and \((8, -4)\). So the vertices are \((2, 8)\), \((14, 8)\), \((14, -4)\), \((8, -4)\).

Step2: Evaluate \(P = 3x - y\) at each point

• For \((8, -4)\): \(P = 3(8)-(-4)=24 + 4 = 28\)
• For \((14, -4)\): \(P = 3(14)-(-4)=42 + 4 = 46\)
• For \((2, 8)\): \(P = 3(2)-8 = 6 - 8 = -2\)
• For \((14, 8)\): \(P = 3(14)-8 = 42 - 8 = 34\)

Step3: Compare values

Compare the \(P\) values: \(46\) (from \((14, -4)\)), \(34\) (from \((14, 8)\)), \(28\) (from \((8, -4)\)), \(-2\) (from \((2, 8)\)). The maximum is \(46\) at \((14, -4)\)? Wait, wait, no, wait the options: the options are \((8, -4)\), \((14, -4)\), \((2, 8)\), \((14, 8)\). Wait, let's recalculate:

Wait, \((14, -4)\): \(3*14 - (-4)=42 + 4 = 46\)

\((14, 8)\): \(3*14 - 8 = 42 - 8 = 34\)

\((8, -4)\): \(3*8 - (-4)=24 + 4 = 28\)

\((2, 8)\): \(3*2 - 8 = -2\)

Wait, but the option \((14, -4)\) is one of the choices? Wait the options given are \((8, -4)\), \((14, -4)\), \((2, 8)\), \((14, 8)\). So when we calculate, \((14, -4)\) gives \(P = 46\), which is the highest. Wait, but let's check the coordinates again. Wait the vertical line is through \((14, 0)\), so \(x = 14\). The horizontal lines: one through \((0, 8)\) (so \(y = 8\)) and one through \((0, -4)\) (so \(y=-4\)). The line through \((2, 8)\) and \((8, -4)\). So the intersection of the line \(x = 14\) with \(y = 8\) is \((14, 8)\), with \(y=-4\) is \((14, -4)\). The other vertices are \((2, 8)\) (intersection of the slant line and \(y = 8\)) and \((8, -4)\) (intersection of the slant line and \(y=-4\)). So the four vertices are \((2, 8)\), \((14, 8)\), \((14, -4)\), \((8, -4)\).

Now, evaluating \(P = 3x - y\) at each:

• \((2, 8)\): \(3*2 - 8 = -2\)
• \((14, 8)\): \(3*14 - 8 = 34\)
• \((14, -4)\): \(3*14 - (-4) = 46\)
• \((8, -4)\): \(3*8 - (-4) = 28\)

So the maximum is at \((14, -4)\). Wait, but let's check the options. The options are:

A. \((8, -4)\)

B. \((14, -4)\)

C. \((2, 8)\)

D. \((14, 8)\)

So the correct one is \((14, -4)\), which is option B (assuming the options are labeled as such, but in the given options, the second option is \((14, -4)\)).

Wait, maybe I made a mistake in the vertices. Wait the line through \((2, 8)\) and \((8, -4)\): let's find where it intersects the vertical line \(x = 14\)? Wait no, the vertical line is at \(x = 14\), and the horizontal lines are at \(y = 8\) and \(y=-4\). So the trapezoid has two parallel sides (the horizontal lines) and the other two sides: one is the vertical line \(x = 14\) (from \(y=-4\) to \(y = 8\)), and the other is the line from \((2, 8)\) to \((8, -4)\). So the vertices are \((2, 8)\) (top left), \((14, 8)\) (top right), \((14, -4)\) (bottom right), \((8, -4)\) (bottom left). Yes, that's correct.

So evaluating \(P = 3x - y\) at each:

• \((2, 8)\): \(3*2 - 8 = -2\)
• \((14, 8)\): \(3*14 - 8 = 34\)
• \((14, -4)\): \(3*14 - (-4) = 46\)
• \((8, -4)\): \(3*8 - (-4) = 28\)

So the maximum is at \((14, -4)\).

Answer:

\(\boldsymbol{(14, -4)}\) (which is the second option, so B. \((14, -4)\))