Quadratic Functions Formulas

Why This Matters

Quadratic functions are the backbone of Algebra 2, and you're being tested on far more than just plugging numbers into formulas. The real exam challenge is knowing when to use each form and why one approach works better than another. Whether you're graphing a parabola, finding roots, or analyzing transformations, the form you choose determines how quickly and accurately you can solve the problem.

Think of these formulas as different lenses for viewing the same function—each one reveals specific information instantly. Standard form shows you the y-intercept, vertex form hands you the vertex coordinates, and factored form displays the roots right in the equation. Master the connections between these forms, and you'll handle any quadratic problem with confidence. Don't just memorize the formulas—know what each one tells you and when it's your best tool.

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The Three Forms of Quadratic Functions

Every quadratic function can be written in three equivalent forms. Each form is optimized to reveal different features of the parabola, so choosing the right form saves time and reduces errors.

Standard Form

• $f(x) = ax^2 + bx + c$—the default form where $a$, $b$, and $c$ are constants and $a \neq 0$
• The y-intercept is $c$—when $x = 0$, you get $f(0) = c$ immediately without calculation
• The coefficient $a$ controls direction—parabola opens upward if $a > 0$, downward if $a < 0$

Vertex Form

• $f(x) = a(x - h)^2 + k$—directly displays the vertex at point $(h, k)$
• Transformations are visible—$h$ shifts horizontally, $k$ shifts vertically, and $a$ stretches or compresses
• Ideal for graphing—plot the vertex first, then use $a$ to determine the parabola's shape and direction

Factored Form

• $f(x) = a(x - r_1)(x - r_2)$—reveals the roots (x-intercepts) at $x = r_1$ and $x = r_2$
• Setting $f(x) = 0$ is trivial—the zero product property gives you roots instantly
• Only works when real roots exist—if the discriminant is negative, this form requires complex numbers

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Finding Roots: The Quadratic Formula and Discriminant

When factoring fails or isn't obvious, these formulas guarantee you can find the roots of any quadratic equation. The discriminant tells you what kind of roots to expect before you even solve.

Quadratic Formula

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—works for every quadratic equation in standard form
• The $\pm$ symbol produces two solutions—add the square root for one root, subtract for the other
• Essential when factoring isn't possible—irrational or complex roots require this formula

Discriminant

• $D = b^2 - 4ac$—the expression under the radical in the quadratic formula
• Determines root type: $D > 0$ means two distinct real roots; $D = 0$ means one repeated real root; $D < 0$ means two complex conjugate roots
• Quick analysis tool—calculate the discriminant first to know what kind of answer to expect

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Locating the Vertex and Axis of Symmetry

These formulas help you find the turning point of any parabola directly from standard form coefficients. The axis of symmetry always passes through the vertex, dividing the parabola into mirror images.

Axis of Symmetry

• $x = \frac{-b}{2a}$—the vertical line that splits the parabola into two symmetric halves
• Also gives the x-coordinate of the vertex—this value is halfway between the two roots
• Critical for graphing—find this line first, then plot symmetric points on either side

Vertex Formula

• Vertex coordinates: $\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$—find $x$ first, then substitute back to get $y$
• Identifies maximum or minimum—if $a > 0$, the vertex is a minimum; if $a < 0$, it's a maximum
• Connects standard form to vertex form—once you find $(h, k)$, you can rewrite in vertex form

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Root Relationships: Vieta's Formulas

These elegant relationships connect the coefficients of a quadratic directly to its roots—no solving required. Vieta's formulas let you analyze roots using only $a$, $b$, and $c$.

Sum of Roots

• $r_1 + r_2 = \frac{-b}{a}$—add the roots without finding them individually
• Reflects the axis of symmetry—the average of the roots equals $\frac{-b}{2a}$, the axis of symmetry
• Useful for checking answers—verify your calculated roots satisfy this relationship

Product of Roots

• $r_1 \cdot r_2 = \frac{c}{a}$—multiply the roots using only coefficients
• Connects to the constant term—when $a = 1$, the product of roots equals $c$
• Helps construct equations—if you know the roots, you can build the quadratic from sum and product

Vieta's Formulas (Combined)

• $r_1 + r_2 = \frac{-b}{a}$ and $r_1 \cdot r_2 = \frac{c}{a}$—the complete relationship between coefficients and roots
• Works for any quadratic—applies whether roots are real, repeated, or complex
• Powerful for "find the equation" problems—given two roots, construct the quadratic without expanding

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Quick Reference Table

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Self-Check Questions

1. You're given a quadratic in standard form and asked to graph it quickly. Which two formulas would you use first to locate the vertex?

2. A quadratic has a discriminant of $-16$. What does this tell you about the roots, and which form of the function would be impossible to write using only real numbers?

3. Compare and contrast: How do vertex form and the vertex formula each help you find the vertex? When would you use one over the other?

4. If you know a quadratic has roots at $x = 3$ and $x = -5$, how would you use Vieta's formulas to find $b$ and $c$ (assuming $a = 1$)?

5. An FRQ gives you $f(x) = 2x^2 - 8x + 6$ and asks for the minimum value. Which formula gives you the x-coordinate of the minimum, and how do you find the actual minimum value?