Discover the beautiful world of parabolas! Learn about vertices, axes of symmetry, and how to transform quadratic functions to create stunning graphs.
• a > 0: Opens upward (∪)
• a < 0: Opens downward (∩)
• |a| > 1: Narrower
• |a| < 1: Wider
Point: (0, c)
x = -b/(2a)
y = f(-b/(2a))
• Vertex: (h, k) - directly visible!
• Axis of Symmetry: x = h
• Transformation: Easy to see shifts and stretches
• Graphing: Start from vertex and plot points
Example: Convert y = 2x² - 8x + 5 to vertex form
Step 1: Factor out the coefficient of x²
y = 2(x² - 4x) + 5
Step 2: Complete the square
y = 2(x² - 4x + 4 - 4) + 5
y = 2((x - 2)² - 4) + 5
Step 3: Distribute and simplify
y = 2(x - 2)² - 8 + 5
y = 2(x - 2)² - 3
Result: Vertex form with vertex at (2, -3)
Use x = -b/(2a) to find x-coordinate, then substitute to find y
Vertical line through the vertex (x = vertex x-coordinate)
Substitute x = 0 to find (0, c)
Set y = 0 and solve (if they exist)
Use symmetry to find points on both sides of the axis
Connect points with a smooth curve
Graph: y = x² - 4x + 3
Step 1: Find vertex
x = -(-4)/(2×1) = 4/2 = 2
y = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1
Vertex: (2, -1)
Step 2: Axis of symmetry
x = 2
Step 3: Y-intercept
y = (0)² - 4(0) + 3 = 3
Y-intercept: (0, 3)
Step 4: X-intercepts
0 = x² - 4x + 3 = (x - 1)(x - 3)
X-intercepts: (1, 0) and (3, 0)
Step 5: Additional points (using symmetry)
Point (0, 3) has symmetric point (4, 3)
Additional point: (4, 3)
Graph Features:
y = (x - h)²: Shift right by h units
y = (x + h)²: Shift left by h units
y = x² + k: Shift up by k units
y = x² - k: Shift down by k units
y = ax² (|a| > 1): Vertical stretch
y = ax² (|a| < 1): Vertical compression
y = -x²: Reflect over x-axis
y = -ax²: Reflect and stretch/compress
Function: y = -2(x - 3)² + 4
Reflection: Negative sign flips parabola downward
Vertical stretch: Factor of 2 makes it narrower
Horizontal shift: (x - 3) shifts right 3 units
Vertical shift: +4 shifts up 4 units
Final vertex: (3, 4)
Remember: x-coordinate of vertex is -b/(2a), not b/(2a)
(x - h) shifts RIGHT, (x + h) shifts LEFT (opposite of what you might expect)
Use the axis of symmetry to find additional points efficiently
Problem 1:
Find the vertex of y = x² - 6x + 8
x = -(-6)/(2×1) = 3
y = (3)² - 6(3) + 8 = 9 - 18 + 8 = -1
Vertex: (3, -1)
Problem 2:
Convert y = 3x² - 12x + 7 to vertex form
y = 3(x² - 4x) + 7
y = 3(x² - 4x + 4 - 4) + 7
y = 3((x - 2)² - 4) + 7
y = 3(x - 2)² - 12 + 7
y = 3(x - 2)² - 5
Problem 3:
Describe the transformations in y = -2(x + 1)² - 3
• Reflection over x-axis (negative sign)
• Vertical stretch by factor of 2
• Horizontal shift left 1 unit
• Vertical shift down 3 units
• Vertex: (-1, -3)