Question 1

We will draw graphs of linear equations. We will observe the roles of 'a' and 'b'.

Step 1 — Graphing $y = ax$ with positive 'a'

Let's plot points for each line. All lines pass through the origin (0,0).

For $y = 4x$: If $x = 0$, then $y = 4(0) = 0$. If $x = 1$, then $y = 4(1) = 4$. Points are (0,0) and (1,4).

For $y = 2x$: If $x = 0$, then $y = 2(0) = 0$. If $x = 1$, then $y = 2(1) = 2$. Points are (0,0) and (1,2).

For $y = x$: If $x = 0$, then $y = 0$. If $x = 1$, then $y = 1$. Points are (0,0) and (1,1).

Let's draw these lines on a graph.

All these lines are of the form $y = ax$. The constant term 'b' is 0.

$\boxed{\text{All lines pass through the origin (0,0).}}$

The value of 'a' is the coefficient of 'x'. It tells us the slope.

$\boxed{\text{A larger positive 'a' means a steeper line.}}$

Step 2 — Graphing $y = ax$ with negative 'a'

Let's plot points for these lines. All lines pass through the origin (0,0).

For $y = -6x$: If $x = 0$, then $y = 0$. If $x = 1$, then $y = -6$. Points are (0,0) and (1,-6).

For $y = -3x$: If $x = 0$, then $y = 0$. If $x = 1$, then $y = -3$. Points are (0,0) and (1,-3).

For $y = -x$: If $x = 0$, then $y = 0$. If $x = 1$, then $y = -1$. Points are (0,0) and (1,-1).

Let's draw these lines on a graph.

All these lines are of the form $y = ax$. The constant term 'b' is 0.

$\boxed{\text{All lines pass through the origin (0,0).}}$

The value of 'a' is negative. This means the lines slope downwards.

$\boxed{\text{A larger absolute value of negative 'a' means a steeper downward slope.}}$

Step 3 — Graphing $y = ax$ and $y = -ax$

Let's plot points for these two lines. Both pass through the origin (0,0).

For $y = 5x$: If $x = 0$, then $y = 0$. If $x = 1$, then $y = 5$. Points are (0,0) and (1,5).

For $y = -5x$: If $x = 0$, then $y = 0$. If $x = 1$, then $y = -5$. Points are (0,0) and (1,-5).

Let's draw these lines on a graph.

Both lines are of the form $y = ax$. The constant term 'b' is 0.

$\boxed{\text{Both lines pass through the origin (0,0).}}$

The 'a' values are 5 and -5. They have the same magnitude.

$\boxed{\text{Same magnitude of 'a' means same steepness but opposite direction.}}$

Step 4 — Graphing $y = ax + b$ with constant 'a'

Let's plot points for each line. The coefficient of 'x' is 3 for all.

For $y = 3x - 1$: If $x = 0$, then $y = 3(0) - 1 = -1$. If $x = 1$, then $y = 3(1) - 1 = 2$. Points are (0,-1) and (1,2).

For $y = 3x$: If $x = 0$, then $y = 3(0) = 0$. If $x = 1$, then $y = 3(1) = 3$. Points are (0,0) and (1,3).

For $y = 3x + 1$: If $x = 0$, then $y = 3(0) + 1 = 1$. If $x = 1$, then $y = 3(1) + 1 = 4$. Points are (0,1) and (1,4).

Let's draw these lines on a graph.

All these lines have the same slope, which is 3.

$\boxed{\text{Lines with the same slope are parallel.}}$

The value of 'b' changes for each line. This is the y-intercept.

$\boxed{\text{The value of 'b' shifts the line vertically.}}$

Step 5 — Graphing $y = ax + b$ with varying 'a' and 'b'

Let's plot points for each line.

For $y = -2x - 3$: If $x = 0$, then $y = -2(0) - 3 = -3$. If $x = 1$, then $y = -2(1) - 3 = -5$. Points are (0,-3) and (1,-5).

For $y = -2x$: If $x = 0$, then $y = -2(0) = 0$. If $x = 1$, then $y = -2(1) = -2$. Points are (0,0) and (1,-2).

For $y = 2x + 3$: If $x = 0$, then $y = 2(0) + 3 = 3$. If $x = 1$, then $y = 2(1) + 3 = 5$. Points are (0,3) and (1,5).

Let's draw these lines on a graph.

Lines $y = -2x - 3$ and $y = -2x$ have the same slope (-2).

$\boxed{\text{Lines with the same slope are parallel.}}$

Line $y = 2x + 3$ has a positive slope (2). It slopes in the opposite direction.

$\boxed{\text{The value of 'b' determines the y-intercept.}}$

Answer

(i) All lines pass through the origin. A larger positive 'a' makes the line steeper. (ii) All lines pass through the origin. A larger absolute value of negative 'a' makes the downward slope steeper. (iii) Both lines pass through the origin. They have the same steepness but opposite directions. (iv) All lines are parallel. The value of 'b' shifts the line up or down. (v) Lines $y = -2x - 3$ and $y = -2x$ are parallel. Line $y = 2x + 3$ slopes in the opposite direction. 'b' sets the y-intercept. Conclusion: The coefficient 'a' (of x) controls the slope. This means it controls steepness and direction. The constant term 'b' controls the vertical shift. This is the y-intercept.