Identifying Transformations
What transformation occurs in y=(x-3)2+5
Shift right 3 units, up 5 units
What is the highest or lowest point on a quadratic function called?
The vertex
What is the vertex of y = x² - 8x + 12? (Hint: Use -b/2a)
Vertex at (4,-4)
What does the vertex of a quadratic function represent in a projectile motion problem?
The highest point (maximum height) of the object
What is the axis of symmetry for y = -2x² + 4x - 1?
x=1
How does the graph of y = -x² compare to y = x²?
Reflected over the x-axis
What is the vertical line that divides a parabola into two symmetrical halves?
Axis of symmetry
Convert y = -2(x - 1)² + 3 to standard form.
y = -2x² + 4x + 1
If a bridge has an arch modeled by y = -x² + 6x + 8, what is the maximum height of the arch?
Maximum height at y = 17
What transformations are needed to shift y = x² so its vertex is at (1, 3)?
Shift right 1, up 3 → equation: y = (x - 1)² + 3
What is the effect of changing y = x² to y = 3x² - 4?
Vertically stretched by 3, shifted down 4
If a quadratic equation has two solutions, what do they represent on the graph?
The x-intercepts, zeros, or roots
Convert y = x² + 6x + 5 to vertex form.
y = (x + 3)² - 4
A business’s profit is modeled by f(x) = -2(x - 10)² + 50. What is the maximum profit?
$50 when x=10
Rewrite y = x² + 8x + 15 in vertex form and describe its transformations.
y=(x + 4)²-1; shift left 4, down 1
How is y = ½(x - 2)² different from y = (x - 2)²?
It is vertically compressed by ½
What does a negative "a" value in y = ax² + bx + c tell us about the graph?
The parabola opens downward
Given y = -x² + 4x - 3, rewrite it in vertex form.
y = -(x - 2)² + 1
The equation for the path of a thrown ball is f(x) = -5(x - 2)² + 20. When does the ball reach its highest point?
At x=2 seconds
If the function y = ax² + bx + c has a minimum at x = 5, what does that tell us about "b"?
b=-10a, since x=-b/2a=5
What transformation happens in y = 2(x + 4)² - 7?
Vertically stretched by a factor of 2, shift left 4, and down 7
What form of a quadratic equation makes it easiest to find the vertex?
Vertex form (y = a(x - h)² + k)
Write an equation in standard form with a vertex at (2, -5) and a stretch factor of 3.
y = 3(x - 2)² - 5, then expand to get y = 3x² - 12x + 7
A diver jumps off a platform, and their height is modeled by h(t) = -4t² + 8t + 16. What is the maximum height the diver reaches?
20 feet
A parabola passes through (0, 10) and has a vertex at (-3, 4). Find its equation in vertex form.
Use y=a(x+3)²+4, then plug in (0,10) to solve for a. Equation: y=(2/3)(x+3)²+4