Key Points to Remember:
- Do NOT plot points
- No need for a scale on the axis (sketched graphs are not accurate)
- The quadratic must be in the form y = ax^2 + bx + c (a cannot equal 0)
- The shape of the curve should be a parabola
Calculating the X Intercepts:
eg: y = x^2 + 2x - 8
Make y = "0" and then solve:
x^2 + 2x - 8 = 0
(X+4)(x-2) = 0
x = -4 or x = 2
Make y = "0" and then solve:
x^2 + 2x - 8 = 0
(X+4)(x-2) = 0
x = -4 or x = 2
Calculating the Y intercept:
eg: y = x^2 + 2x - 8
Make x = "0"
Therefore y = c (as 0^2 and b(0) cancel out)
y = -8
Make x = "0"
Therefore y = c (as 0^2 and b(0) cancel out)
y = -8
Calculating the Line of Symmetry:
- The Line of Symmetry crosses the x-axis halfway between the two X-Intercepts.
- Therefore: (-4+2)/2 = -1
- So the equation of the Line of Symmetry is: x = -1
Calculating the Coordinates of the Vertex:
- The X coordinate of the Vertex is the same as the line of Symmetry
- Therefore x = -1 is the same as (-1, y)
- To work out the Y coordinate of the Vertex, substitute the X coordinate into the equation of the curve
- Therefore y = (-1)^2 + 2(-1) - 8
- Therefore y = -9
- Therefore the coordinates of the Vertex are (-1, -9)
- When completing the square in the form (x+p)^2 + q, the coordinates of the Vertex would be (-p, q)
Sketching the Graph Itself:
- Draw a curve correspondent to the x value (positive curve/negative curve)
- Draw the x-axis so it crosses the parabola twice
- Mark your X-Intercepts on the x axis
- Draw your y axis relative to the x intercepts
- Mark your Y-Intercept on the y axis
- Draw on the Line of Symmetry as a dotted line and state it's equation
- Mark on the vertex and its coordinates
Worked Example 1:
A quadratic curve has the equation: y = x^2 + 4x + 3
a) Find the Y-Intercept y = +3 b) Find the X-Intercepts x^2 + 4x + 3 = 0 (x+3)(x+1) = o x = -3 or x = -1 c) Find the Line of Symmetry (-3+-1)/2 = -2 Equation of the Line of Symmetry = x = -2 d)Find the coordinates of the Vertex X coordinate = -2 y = (-2)^2 + 4(-2) + 3 y = -1 Vertex = (-2, -1) e) Sketch the graph of y = x^2 + 4x + 3 Graph to the right >>> |
Worked Example 2:
A quadratic curve has the equation: y = x^2 - 36
a) Find the Y-Intercept y = -36 b) Find the X-Intercepts x^2 - 36 = 0 (x+6)(x-6) = o x = -6 or x = 6 c) Find the Line of Symmetry (-6+6)/2 = 0 Equation of the Line of Symmetry = x = 0 d)Find the coordinates of the Vertex X coordinate = 0 y = (0)^2 -36 y = -36 Vertex = (0, -36) e) Sketch the graph of y = x^2 - 36 Graph to the right >>> |
Worked Example 3:
A quadratic curve has the equation: y = x^2+3x -10
a) Find the Y-Intercept y = -10 b) Find the X-Intercepts x^2 + 3x - 10 = 0 (x+5)(x-2) = o x = -5 or x = 2 c) Find the Line of Symmetry (-5+2)/2 = -3/2 Equation of the Line of Symmetry = x = -3/2 d)Find the coordinates of the Vertex X coordinate = -3/2 y = (-3/2)^2 + 3(-3/2) - 10 y = 9/4 - 9/2 - 10 y = (9-18-40)/4 y = -49/4 Vertex = (-3/2, -49/4) e) Sketch the graph of y = x^2 + 3x - 10 Graph to the right >>> |