Quadratic Standard Form Notes
Standard Form of Quadratic Functions
Identifying Coefficients \(a\), \(b\), and \(c\)
\(a\) is with \(x^2\) • \(b\) is with \(x\) • \(c\) is the constant
| Given Equation | Standard Order | \(a\) | \(b\) | \(c\) | Note |
|---|
| \(y = 2x^2 - 5x + 3\) | \(2x^2 - 5x + 3\) | \(2\) | \(-5\) | \(3\) | Already in standard order; negative sign belongs to \(-5\) |
| \(y = 6 - 4x^2 + 9x\) | | | | | \(a\) is with \(x^2\), \(b\) with \(x\), and \(c\) is the constant |
| \(y = 7x - x^2\) | | | | | Missing constant term means \(c = 0\); invisible coefficient is \(-1\) |
Direction of Opening & The \(y\)-Intercept
The direction of opening is determined by whether the leading coefficient \(a\) is positive or negative:
If \(a\) is positive → opens up
If \(a\) is negative → opens down
\(y\)-Intercept
\(x = 0\) → \((0, c)\)
Practice: Identify Features from Given Equations
| Given Equation | Standard Order | Sign of \(a\) & Direction of Opening | \(y\)-Intercept \((0, c)\) |
|---|
| \(y = -3x^2 + 8x - 5\) | | | |
| \(y = 4 - x^2 + 6x\) | | | |
| \(y = 10x - 2x^2\) | | | |
Introducing the Axis of Symmetry
The axis of symmetry is the vertical line dividing the parabola into mirrored halves. Always write as an equation:
Axis of Symmetry: \(x = -\frac{b}{2a}\)
Solved Example
\(y = 12x - 3x^2 - 7\)
1. Standard Order: \(y = -3x^2 + 12x - 7 \implies a = -3,\; b = 12,\; c = -7\)
2. Apply formula:
\(x = -\frac{b}{2a} = -\frac{12}{2(-3)} = -\frac{12}{-6} = 2\)
Axis of Symmetry: \(x = 2\)
Solve for Axis of Symmetry
\(y = 4 - 8x - 2x^2\)
1. Standard Order:
2. Apply formula:
Axis of Symmetry:
Solving for the Vertex \((h, k)\)
Step 1 (\(x\)-value): \(h\) is equal to the value from the axis of symmetry equation: \(x = h = -\frac{b}{2a}\). Step 2 (\(y\)-value): Substitute \(h\) into the original equation to solve for \(y\).
Solved Example
\(y = 6x - x^2 + 1\)
1. Standard Order: \(y = -x^2 + 6x + 1 \implies a = -1,\; b = 6,\; c = 1\)
2. Solve for \(x\) (\(h\)):
\(x = -\frac{6}{2(-1)} = -\frac{6}{-2} = 3\)
3. Solve for \(y\) (from original equation):
\(y = 6(3) - (3)^2 + 1 = 18 - 9 + 1 = 10\)
Vertex: \((3, 10)\) ( max / min )
Solve for Vertex
\(y = 3 - 8x + 2x^2\)
1. Standard Order:
2. Solve for \(x\) (\(h\)):
3. Solve for \(y\) (from original equation):
Vertex: ( max / min )
Standard Form Quadratic Practice
Standard Form of Quadratic Equations
Name:
Directions: For each quadratic equation, identify the \(y\)-intercept, calculate the axis of symmetry equation (\(x = -\frac{b}{2a}\)), and determine the vertex coordinates \((h, k)\). Circle whether the vertex is a maximum or minimum.
Problem 1
\(y = x^2 - 6x + 5\)
Standard Order:
Axis of Symmetry Work:
Vertex Work:
\(y\)-Intercept:
(0, y)
Axis of Symmetry:
x =
Vertex:
(h, k)
( max / min )
Problem 2
\(y = -2x^2 + 8x - 3\)
Standard Order:
Axis of Symmetry Work:
Vertex Work:
\(y\)-Intercept:
(0, y)
Axis of Symmetry:
x =
Vertex:
(h, k)
( max / min )
Problem 3
\(y = 2x^2 - 6x + 1\)
Standard Order:
Axis of Symmetry Work:
Vertex Work:
\(y\)-Intercept:
(0, y)
Axis of Symmetry:
x =
Vertex:
(h, k)
( max / min )
Problem 4
\(y = 5 + 4x - \frac{1}{2}x^2\)
Standard Order:
Axis of Symmetry Work:
Vertex Work:
\(y\)-Intercept:
Axis of Symmetry:
Vertex:
( max / min )
Problems 5 – 8 on back
Problem 5
\(y = 10x - 2x^2 - 7\)
Standard Order:
Axis of Symmetry Work:
Vertex Work:
\(y\)-Intercept:
Axis of Symmetry:
Vertex:
( max / min )
Problem 6
\(y = \frac{1}{3}x^2 - 4 - 2x\)
Standard Order:
Axis of Symmetry Work:
Vertex Work:
\(y\)-Intercept:
Axis of Symmetry:
Vertex:
( max / min )
Problem 7
\(y = 4x - 4x^2 + 3\)
\(y\)-Intercept:
Axis of Symmetry:
Vertex:
( max / min )
Problem 8
\(y = 2x - \frac{1}{4}x^2\)
\(y\)-Intercept:
Axis of Symmetry:
Vertex:
