Vertex Calculator
Convert quadratic functions to vertex form and find the vertex with step-by-step solutions
Standard Form: \(f(x) = x^2\)
What is Vertex Form?
The vertex form of a quadratic function is:
\(f(x) = a(x - h)^2 + k\)
where \((h, k)\) is the vertex of the parabola and \(a\) determines the width and direction of the parabola:
- If \(a > 0\), the parabola opens upward
- If \(a < 0\), the parabola opens downward
- The larger the absolute value of \(a\), the narrower the parabola
Converting to Vertex Form
To convert from standard form \(ax^2 + bx + c\) to vertex form \(a(x - h)^2 + k\):
- Find the x-coordinate of the vertex: \(h = -\frac{b}{2a}\)
- Find the y-coordinate of the vertex: \(k = f(h) = a \cdot h^2 + b \cdot h + c\)
- Substitute these values into the vertex form equation
This process is called "completing the square."
Why Use Vertex Form?
Vertex form is useful because it:
- Immediately tells you the vertex (maximum or minimum point) of the parabola
- Makes it easy to graph the parabola
- Simplifies finding the axis of symmetry
- Helps in solving optimization problems
- Makes transformations of the function more intuitive
Examples
Basic Example
Convert \(f(x) = x^2 + 6x + 8\) to vertex form
Step 1: Find the x-coordinate of the vertex
\(h = -\frac{b}{2a} = -\frac{6}{2 \cdot 1} = -3\)
Step 2: Find the y-coordinate of the vertex
\(k = f(-3) = 1 \cdot (-3)^2 + 6 \cdot (-3) + 8\)
\(= 9 - 18 + 8 = -1\)
Step 3: Write in vertex form
\(f(x) = 1 \cdot (x - (-3))^2 + (-1)\)
\(= (x + 3)^2 - 1\)
Negative Leading Coefficient
Convert \(f(x) = -2x^2 + 8x - 3\) to vertex form
Step 1: Find the x-coordinate of the vertex
\(h = -\frac{b}{2a} = -\frac{8}{2 \cdot (-2)} = -\frac{8}{-4} = 2\)
Step 2: Find the y-coordinate of the vertex
\(k = f(2) = -2 \cdot (2)^2 + 8 \cdot (2) - 3\)
\(= -8 + 16 - 3 = 5\)
Step 3: Write in vertex form
\(f(x) = -2 \cdot (x - 2)^2 + 5\)
Completing the Square Method
Convert \(f(x) = 3x^2 - 12x + 7\) to vertex form
Step 1: Factor out the leading coefficient
\(f(x) = 3(x^2 - 4x) + 7\)
Step 2: Complete the square inside the parentheses
Take half the coefficient of x and square it: \((-4/2)^2 = (-2)^2 = 4\)
\(f(x) = 3(x^2 - 4x + 4 - 4) + 7\)
\(f(x) = 3((x - 2)^2 - 4) + 7\)
\(f(x) = 3(x - 2)^2 - 12 + 7\)
\(f(x) = 3(x - 2)^2 - 5\)
Frequently Asked Questions
What is the vertex of a parabola?
The vertex of a parabola is the highest or lowest point on the graph. For a parabola that opens upward (a > 0), the vertex is the minimum point. For a parabola that opens downward (a < 0), the vertex is the maximum point. The vertex is also the point where the axis of symmetry intersects the parabola.
What's the difference between standard form and vertex form?
Standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. This form is useful for finding x-intercepts and evaluating the function at specific x-values.
Vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex, determine the direction the parabola opens, and graph the function.
Can I convert from vertex form back to standard form?
Yes, you can convert from vertex form f(x) = a(x - h)² + k back to standard form by expanding the expression:
f(x) = a(x - h)² + k
f(x) = a(x² - 2hx + h²) + k
f(x) = ax² - 2ahx + ah² + k
This gives you the standard form with coefficients:
a = a
b = -2ah
c = ah² + k
Applications of Vertex Calculation
Projectile Motion
The vertex of a projectile's parabolic trajectory represents its maximum height. Vertex form helps calculate:
- Peak height of thrown objects
- Time to reach maximum height
- Optimal launch angles
Business Optimization
Vertex calculations help businesses find:
- Maximum profit points
- Optimal pricing strategies
- Minimum cost production levels
Engineering Design
Engineers use vertex form for:
- Designing parabolic arches and bridges
- Calculating optimal structural loads
- Determining focal points in satellite dishes