Graphing Linear and Quadratic Functions

Graphing Linear Equations

• Graph of Linear Equations: To graph linear equations of the form $kx + y = b$, find at least two points that satisfy the equation and draw a straight line through them.

1. Equation: $2x + y = 12$

• Point 1: Set $x = 0$:
  • $2(0) + y = 12$
  • $y = 12$ (Point: (0, 12))
• Point 2: Set $y = 0$:
  • $2x + 0 = 12$
  • $x = 6$ (Point: (6, 0))
• Graph: Plot the points (0, 12) and (6, 0), and draw a line through them.

2. Equation: $3x + y = 12$

• Point 1: Set $x = 0$:
  • $3(0) + y = 12$
  • $y = 12$ (Point: (0, 12))
• Point 2: Set $y = 0$:
  • $3x + 0 = 12$
  • $x = 4$ (Point: (4, 0))
• Graph: Plot the points (0, 12) and (4, 0), and draw a line through them.

3. Equation: $4x + y = 12$

• Point 1: Set $x = 0$:
  • $4(0) + y = 12$
  • $y = 12$ (Point: (0, 12))
• Point 2: Set $y = 0$:
  • $4x + 0 = 12$
  • $x = 3$ (Point: (3, 0))
• Graph: Plot the points (0, 12) and (3, 0), and draw a line through them.

4. General Case: Equation: $kx + y = 12$

• Description:
  • For varying values of $k$, the slope of the line changes, while the y-intercept remains constant at 12.
  • A larger absolute value of $k$ indicates a steeper slope.

5. Equation: $x + 2y = 14$

• Point 1: Set $x = 0$:
  • $0 + 2y = 14$
  • $y = 7$ (Point: (0, 7))
• Point 2: Set $y = 0$:
  • $x + 0 = 14$
  • $x = 14$ (Point: (14, 0))
• Graph: Plot the points (0, 7) and (14, 0), and draw a line through them.

6. General Case: Equation: $x + 2y = n$ when $n$ varies

• Description:
  • The slope of the line is $-1/2$, and the y-intercept will change depending on the value of $n$.
  • The intercept can be found by setting $x = 0$, which gives a y-intercept of $y = n/2$.

Graphing Quadratic Equations

1. Equation: $y = x^2$

• The graph is a parabola opening upwards with its vertex at the origin (0,0).
• Table of Values:
  • For different $x$ values, calculate corresponding $y$:
  • $x = -3, y = 9$
  • $x = -2, y = 4$
  • $x = -1, y = 1$
  • $x = 0, y = 0$
  • $x = 1, y = 1$
  • $x = 2, y = 4$
  • $x = 3, y = 9$
• Graph: Plot the points and draw a curve.

2. Equation: $y = x^2 + 2$

• The graph is similar to $y = x^2$, but shifted upwards by 2 units.
• Table of Values:
  • For different $x$ values:
  • $x = -3, y = 11$
  • $x = -2, y = 6$
  • $x = -1, y = 3$
  • $x = 0, y = 2$
  • $x = 1, y = 3$
  • $x = 2, y = 6$
  • $x = 3, y = 11$
• Graph: Plot the points and draw a curve shifted vertically.

3. Equation: $y = x^2 - 4$

• The graph resembles a standard parabola opening upwards but is shifted downwards by 4 units.
• Table of Values:
  • For different $x$ values:
  • $x = -3, y = 5$
  • $x = -2, y = 0$
  • $x = -1, y = -3$
  • $x = 0, y = -4$
  • $x = 1, y = -3$
  • $x = 2, y = 0$
  • $x = 3, y = 5$
• Graph: Plot the points and draw the upward-opening parabola shifted downward.

4. General Case: Equation: $y = x^2 + c$

• Description:
  • The value of $c$ determines the vertical shift of the parabola.
  • If c > 0, the graph shifts upward; if c < 0, it shifts downward.
• The vertex remains at $(0, c)$ regardless of $c$.

Conclusion

• Understanding how to plot linear and quadratic equations is fundamental to graphical analysis in mathematics. The manipulation of constants in equations affects the slopes and positions of the graphs significantly, enabling complex analyses and applications in higher mathematics.