Math Parabola

VERTEX FORM of a Quadratic Equation

Definition of Vertex Form

• The vertex form of a quadratic equation is expressed as: $y = a(x - h)^2 + k$ where:

  • $a$ is the coefficient that indicates the direction and width of the parabola. If a > 0, the parabola opens upwards; if a < 0, the parabola opens downwards.

  • $h$ is the horizontal shift (left/right) of the parabola.

  • $k$ is the vertical shift (up/down) of the parabola.

  • The vertex of the parabola is given by the point $(h, k)$.

Key Characteristics

• Axis of Symmetry: The axis of symmetry of the parabola is given by the equation:
  $x = h$

Examples of Finding the Axis of Symmetry and Vertex

1. For the equation $y = ext{sqrt}((x + 4)^2) + 2$:

   • Axis of Symmetry: $x = -4$

   • Vertex: $( -4, 2 )$

2. For the equation $y = - (x - 3)^2 - 10$:

   • Axis of Symmetry: $x = 3$

   • Vertex: $(3, -10)$

3. For the equation $y = (x - 5)^2 - 4$:

   • Axis of Symmetry: $x = 5$

   • Vertex: $(5, -4)$

4. For the equation $y = -2x^2 + 3$:

   • Axis of Symmetry: $x = 0$

   • Vertex: $(0, 3)$

Graphing from Vertex Form

1. Identify key components:

   • Determine the vertex, the axis of symmetry, domain, and range of the quadratic equations.

   • Domain for all quadratic functions is:

   • All real numbers: $(- ext{∞}, ext{∞})$

2. Range:

   • For a parabola opening upwards, the range is $[k, ext{∞})$.

   • For a parabola opening downwards, the range is $(- ext{∞}, k]$.

Steps to Graph a Quadratic Equation

1. Identify the axis of symmetry.

2. Determine the vertex.

3. Create a table of values utilizing the vertex as a center point.

4. Use a calculator to compute further points if necessary.

5. Plot the points and connect them with a smooth parabolic curve.

Additional Examples for Graphing Quadratic Equations

1. For $y = x^2 - 1$:

   • Axis of Symmetry: $x = 0$

   • Vertex: $(0, -1)$

2. For $y = x^2 + 2x - 1$:

   • Axis of Symmetry: $x = -1$

   • Vertex: $( -1, -2 )$

3. For $y = -x^2 - 8x - 17$:

   • Axis of Symmetry: $x = -4$

   • Vertex: $( -4, -1 )$

Transformations of Quadratic Functions

• Transformations describe the effect of the parameters $h$, $k$, and $a$ in the vertex form equation:

  • Shifts to the right by $h$ units and left by $-h$ units.

  • Shifts upwards by $k$ units and downwards by $-k$ units.

  • A negative $a$ reflects the graph over the x-axis.

  • Changes in the absolute value of $a$ (e.g., if |a| > 1 or 0 < |a| < 1) describe a vertical stretch or compression, respectively.

Examples of Transformations

1. For the transformation translating 2 units right and 3 units down, the equation becomes:
   $y = (x - 2)^2 - 3$

2. For a vertical stretch by a factor of 2, reflecting over the x-axis, then translating 4 units left, it results in:
   $y = -2(x + 4)^2$

3. For vertical compression of $rac{1}{3}$, then translated 8 units up:
   $y = rac{1}{3}x^2 + 8$

Composing Quadratic Equations from Transformations

• Given the transformations listed:

1. Translated 3 units left and 4 down:

   • Equation: $y = (x + 3)^2 - 4$

2. Reflected over the x-axis, then translated 5 units right and 2 units down:

   • Equation: $y = -(x - 5)^2 - 2$

3. Vertically stretched by a factor of 2, reflected across x-axis:

   • Equation: $y = -2(x + 4)^2$

Overall Notes

• Understanding how to convert between forms and graph quadratic equations is essential for advanced topics in Algebra, Calculus, and further mathematical studies. Transformation properties pave the way for understanding the characteristics of parabolic functions.