Introduction to Graphs

• (a) $P(3, -5)$: Abscissa is positive ($+3$) and ordinate is negative ($-5$) $\implies$ Quadrant IV.
• (b) $Q(-4, -7)$: Both abscissa and ordinate are negative $\implies$ Quadrant III.
• (c) $R(0, 8)$: Abscissa is $0$, ordinate is positive $\implies$ Positive $y$-axis.
• (d) $S(-6, 0)$: Ordinate is $0$, abscissa is negative $\implies$ Negative $x$-axis.
• (e) $T(-2, 9)$: Abscissa is negative and ordinate is positive $\implies$ Quadrant II.

Step 1: Construct the table of values using $\text{Distance} = \text{Speed} \times \text{Time}$ ($d = 60t$):
• $t = 0\text{ h} \implies d = 0\text{ km}$
• $t = 1\text{ h} \implies d = 60\text{ km}$
• $t = 2\text{ h} \implies d = 120\text{ km}$
• $t = 3\text{ h} \implies d = 180\text{ km}$
• $t = 4\text{ h} \implies d = 240\text{ km}$
• $t = 5\text{ h} \implies d = 300\text{ km}$

Step 2: Origin Check: At $t = 0$, distance $d = 0$. Therefore, YES, the graph passes directly through the origin $(0, 0)$.
Step 3: Slope Calculation:

$$\text{Slope} = \frac{\Delta y}{\Delta x} = \frac{120 - 60}{2 - 1} = \mathbf{60\text{ km/h}}$$

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The slope represents the constant uniform speed of the car.

• The formula for the Area of a square of side $s$ is: $A = s^2$.
• Let us test values:
• Side $s = 1 \implies A = 1$
• Side $s = 2 \implies A = 4$ (Change $= 3$)
• Side $s = 3 \implies A = 9$ (Change $= 5$)
• Side $s = 4 \implies A = 16$ (Change $= 7$)
• The rate of change $\frac{\Delta A}{\Delta s}$ is not constant; it increases rapidly.
• When plotted, this produces a curved parabola, not a straight line.
• Therefore, NO, it is NOT a linear graph.

• The linear relationship is: $y = 80x$ (Direct proportion passing through $(0, 0)$).
• For $x = 4.5\text{ kg}$:

$$y = 80 \times 4.5 = 80 \times \frac{9}{2} = 40 \times 9 = \mathbf{\text{Rs } 360}$$

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The cost of $4.5\text{ kg}$ of apples is $\text{Rs } 360$.

• (a) Distance from $x$-axis: The perpendicular distance of any point $(x, y)$ from the $x$-axis is given by the absolute value of its $y$-coordinate (ordinate):

$$\text{Distance from } x\text{-axis} = |y| = |7| = \mathbf{7\text{ units}}$$

• (b) Distance from $y$-axis: The perpendicular distance of any point $(x, y)$ from the $y$-axis is given by the absolute value of its $x$-coordinate (abscissa):

$$\text{Distance from } y\text{-axis} = |x| = |-5| = \mathbf{5\text{ units}}$$

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• A horizontal flat line parallel to the time axis means that as time advances ($x$ increases), the distance value ($y$) remains entirely unchanged ($\Delta y = 0$).
• Therefore:

$$\text{Speed} = \frac{\Delta y}{\Delta x} = \frac{0}{\Delta x} = \mathbf{0}$$

• It signifies that the object is completely stationary (at rest).

• (a) Intersection with $x$-axis: Set $y = 0$ in the equation:

$$2x + 3(0) = 12 \implies 2x = 12 \implies x = 6$$

Coordinates: $(6, 0)$.

• (b) Intersection with $y$-axis: Set $x = 0$ in the equation:

$$2(0) + 3y = 12 \implies 3y = 12 \implies y = 4$$

Coordinates: $(0, 4)$.

• Line Graph: A series of data points connected by consecutive straight line segments. The overall graph may bend, zigzag, rise and fall (non-linear overall, e.g., weekly stock prices or temperature trends).
• Linear Graph: The entire graph consists of one single, unbroken continuous straight line across its entire domain, indicating a constant rate of change ($y = mx + c$).