Question

1. a line passes through a(-3, 1). for each slope given below: i) sketch the line through a with that slope. ii) write the coordinates of three other points on the line. a) -1 b) \\(\frac{1}{4}\\) c) \\(-\frac{3}{2}\\)

Explanation:

Part a) Slope -1

Step1: Recall slope formula

Slope \( m=\frac{y_2 - y_1}{x_2 - x_1}\), given \( A(-3,1)\) and \( m=-1\). Let \((x_1,y_1)=(-3,1)\) and \((x_2,y_2)=(x,y)\). So \(-1=\frac{y - 1}{x+ 3}\), which implies \(y-1=-(x + 3)\) or \(y=-x - 2\).

Step2: Find other points

• For \(x=-2\), \(y=-(-2)-2=0\), so point \((-2,0)\).
• For \(x=-1\), \(y=-(-1)-2=-1\), so point \((-1,-1)\).
• For \(x = 0\), \(y=-0 - 2=-2\), so point \((0,-2)\).

Part b) Slope \(\frac{1}{4}\)

Step1: Recall slope formula

\(m = \frac{1}{4}=\frac{y - 1}{x + 3}\), so \(y-1=\frac{1}{4}(x + 3)\) or \(y=\frac{1}{4}x+\frac{3}{4}+1=\frac{1}{4}x+\frac{7}{4}\).

Step2: Find other points

• For \(x = 1\), \(y=\frac{1}{4}(1)+\frac{7}{4}=\frac{8}{4}=2\), point \((1,2)\).
• For \(x = 5\), \(y=\frac{1}{4}(5)+\frac{7}{4}=\frac{12}{4}=3\), point \((5,3)\).
• For \(x=-7\), \(y=\frac{1}{4}(-7)+\frac{7}{4}=0\), point \((-7,0)\).

Part c) Slope \(-\frac{3}{2}\)

Step1: Recall slope formula

\(m=-\frac{3}{2}=\frac{y - 1}{x + 3}\), so \(y - 1=-\frac{3}{2}(x + 3)\) or \(y=-\frac{3}{2}x-\frac{9}{2}+1=-\frac{3}{2}x-\frac{7}{2}\).

Step2: Find other points

• For \(x=-1\), \(y=-\frac{3}{2}(-1)-\frac{7}{2}=\frac{3 - 7}{2}=-2\), point \((-1,-2)\).
• For \(x = 1\), \(y=-\frac{3}{2}(1)-\frac{7}{2}=\frac{-3 - 7}{2}=-5\), point \((1,-5)\).
• For \(x = 3\), \(y=-\frac{3}{2}(3)-\frac{7}{2}=\frac{-9 - 7}{2}=-8\), point \((3,-8)\).

Sketching (General Idea)

• For slope -1: Start at \((-3,1)\), move 1 unit right and 1 unit down (or 1 unit left and 1 unit up) to plot other points and draw the line.
• For slope \(\frac{1}{4}\): Start at \((-3,1)\), move 4 units right and 1 unit up (or 4 units left and 1 unit down) to plot other points and draw the line.
• For slope \(-\frac{3}{2}\): Start at \((-3,1)\), move 2 units right and 3 units down (or 2 units left and 3 units up) to plot other points and draw the line.

Answer:

s:

Part a)

i) Sketch: Line through \((-3,1)\) with slope -1 (decreasing, 45 - degree angle - like).
ii) Three points: \((-2,0)\), \((-1,-1)\), \((0,-2)\)

Part b)

i) Sketch: Line through \((-3,1)\) with positive, gentle slope.
ii) Three points: \((1,2)\), \((5,3)\), \((-7,0)\)

Part c)

i) Sketch: Line through \((-3,1)\) with negative, steeper slope.
ii) Three points: \((-1,-2)\), \((1,-5)\), \((3,-8)\)