Chapter 10: Graphs of Functions

Solve · page 194: 1. Sketch the graph of the following linear functions:

(i) $y = 3x - 5$ [حخطی افعال کا گراف]

<svg viewBox="0 0 340 240" font-size="13"> <defs> <marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="currentColor"/> </marker> </defs> <!-- Axes --> <line x1="60" y1="200" x2="310" y2="200" stroke="currentColor" marker-end="url(#arrow)" /> <line x1="60" y1="220" x2="60" y2="20" stroke="currentColor" marker-end="url(#arrow)" /> <text x="315" y="204" fill="currentColor">+x</text> <text x="50" y="15" fill="currentColor">+y</text> <text x="52" y="212" fill="currentColor">O</text> <!-- Graph line for y = 3x - 5: points (1,-2) -> (60+20, 200-20*(-2))? Let's map x=1 -> 60+30=90, y=-2 -> 200 - (-2)*30 = 260 off screen. Let's use scale: 1 unit = 20px. Origin (140, 140). x = 0 -> y = -5 -> (140, 140 - (-5)*20) = (140, 240) off screen. Let's choose origin at (140, 100) and scale 1 unit = 20px. x=0, y=-5 -> pixel (140, 100 - (-5)*20) = (140, 200). x=1, y=-2 -> pixel (140 + 20, 100 - (-2)*20) = (160, 140). x=2, y=1 -> pixel (140 + 40, 100 - 1*20) = (180, 80). x=3, y=4 -> pixel (140 + 60, 100 - 4*20) = (200, 20). Let's redraw axes with origin at (140, 160): x=0, y=-5 -> (140, 160 - (-5)*15) = (140, 235) Let's use a standard coordinate mapping for x from 0 to 3, y from -5 to 4. Origin (80, 180), scale x: 60px per unit, y: 20px per unit. x=0, y=-5 -> (80, 180 - (-5)*20) = (80, 280) - too low. Let's use origin (80, 110), scale x=60, y=15: x=0, y=-5 -> (80, 110 - (-5)*15) = (80, 185) x=1, y=-2 -> (80 + 60, 110 - (-2)*15) = (140, 140) x=2, y=1 -> (80 + 120, 110 - 1*15) = (200, 95) x=3, y=4 -> (80 + 180, 110 - 4*15) = (260, 50) --> <!-- Clear axes with origin (80, 110) --> <line x1="80" y1="185" x2="310" y2="185" stroke="currentColor" marker-end="url(#arrow)" /> <line x1="80" y1="210" x2="80" y2="20" stroke="currentColor" marker-end="url(#arrow)" /> <text x="315" y="189" fill="currentColor">+x</text> <text x="70" y="15" fill="currentColor">+y</text> <text x="65" y="197" fill="currentColor">O</text>

<!-- Line --> <line x1="80" y1="185" x2="280" y2="35" stroke="currentColor" stroke-width="2" />

<!-- Points --> <circle cx="140" cy="140" r="3" fill="currentColor" /> <circle cx="200" cy="95" r="3" fill="currentColor" /> <circle cx="260" cy="50" r="3" fill="currentColor" />

<text x="145" y="155" fill="currentColor">(1, -2)</text> <text x="205" y="110" fill="currentColor">(2, 1)</text> <text x="235" y="45" fill="currentColor">y = 3x - 5</text> </svg>

• Scale: X-axis $1 \text{ unit} = 60\text{ px}$, Y-axis $1 \text{ unit} = 15\text{ px}$.
• Construct a table of values by substituting $x = 1, 2, 3$ to get $y = -2, 1, 4$.
• Plot the points on the Cartesian plane and join them with a straight line.

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(ii) $y = -2x + 8$ [حخطی افعال کا گراف]

<svg viewBox="0 0 340 240" font-size="13"> <defs> <marker id="arrow2" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="currentColor"/> </marker> </defs> <!-- Axes with origin (60, 200) and scale x=40, y=15 --> <!-- x=0, y=8 -> (60, 200 - 8*15) = (60, 80) --> <!-- x=2, y=4 -> (60 + 2*40, 200 - 4*15) = (140, 140) --> <!-- x=4, y=0 -> (60 + 4*40, 200 - 0) = (220, 200) --> <line x1="60" y1="200" x2="310" y2="200" stroke="currentColor" marker-end="url(#arrow2)" /> <line x1="60" y1="220" x2="60" y2="20" stroke="currentColor" marker-end="url(#arrow2)" /> <text x="315" y="204" fill="currentColor">+x</text> <text x="50" y="15" fill="currentColor">+y</text> <text x="45" y="212" fill="currentColor">O</text>

<!-- Line --> <line x1="60" y1="80" x2="260" y2="230" stroke="currentColor" stroke-width="2" />

<!-- Points --> <circle cx="60" cy="80" r="3" fill="currentColor" /> <circle cx="140" cy="140" r="3" fill="currentColor" /> <circle cx="220" cy="200" r="3" fill="currentColor" />

<text x="70" y="80" fill="currentColor">(0, 8)</text> <text x="145" y="135" fill="currentColor">(2, 4)</text> <text x="210" y="185" fill="currentColor">(4, 0)</text> <text x="180" y="70" fill="currentColor">y = -2x + 8</text> </svg>

• Scale: X-axis $1 \text{ unit} = 40\text{ px}$, Y-axis $1 \text{ unit} = 15\text{ px}$.
• Construct a table of values by substituting $x = 0, 2, 4$ to get $y = 8, 4, 0$.
• Plot the points on the Cartesian plane and join them with a straight line.

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(iii) $y = 0.5x - 1$ [حخطی افعال کا گراف]

<svg viewBox="0 0 340 240" font-size="13"> <defs> <marker id="arrow3" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse"> <path d="M 0 0 L 10 5 L 0 10 z" fill="currentColor"/> </marker> </defs> <!-- Axes with origin (80, 120) and scale x=50, y=50 --> <!-- x=0, y=-1 -> (80, 120 - (-1)*50) = (80, 170) --> <!-- x=2, y=0 -> (80 + 2*50, 120 - 0) = (180, 120) --> <!-- x=4, y=1 -> (80 + 4*50, 120 - 1*50) = (280, 70) --> <line x1="80" y1="170" x2="310" y2="170" stroke="currentColor" marker-end="url(#arrow3)" /> <line x1="80" y1="210" x2="80" y2="20" stroke="currentColor" marker-end="url(#arrow3)" /> <text x="315" y="174" fill="currentColor">+x</text> <text x="70" y="15" fill="currentColor">+y</text> <text x="65" y="182" fill="currentColor">O</text>

<!-- Line --> <line x1="30" y1="195" x2="310" y2="55" stroke="currentColor" stroke-width="2" />

<!-- Points --> <circle cx="80" cy="170" r="3" fill="currentColor" /> <circle cx="180" cy="120" r="3" fill="currentColor" /> <circle cx="280" cy="70" r="3" fill="currentColor" />

<text x="90" y="170" fill="currentColor">(0, -1)</text> <text x="185" y="115" fill="currentColor">(2, 0)</text> <text x="210" y="65" fill="currentColor">y = 0.5x - 1</text> </svg>

• Scale: X-axis $1 \text{ unit} = 50\text{ px}$, Y-axis $1 \text{ unit} = 50\text{ px}$.
• Construct a table of values by substituting $x = 0, 2, 4$ to get $y = -1, 0, 1$.
• Plot the points on the Cartesian plane and join them with a straight line.