Quadratic Formula Calculator — Solve ax² + bx + c = 0
Solve any quadratic equation with the quadratic formula. Find both roots, discriminant, vertex, axis of symmetry, and factored form with step-by-step solutions.
Y-Intercept Calculator
Find the y-intercept of linear, quadratic, cubic, exponential, and logarithmic functions. Also shows x-intercepts, slope at origin, domain, and range with visual comparison bars.
Y-Intercept⧉
(0, 7.0000)
Point where the graph crosses the y-axis (x = 0)
X-Intercept(s)⧉
x = -2.3333
Point(s) where the graph crosses the x-axis (y = 0)
Slope at Origin⧉
3.0000
Value of f′(0) — instantaneous rate of change at x = 0
Equation⧉
y = 3x + 7
Increasing line
Domain⧉
(−∞, ∞)
All valid input values for x
Range⧉
(−∞, ∞)
All possible output values for y
Intercept Comparison
| Property | Value | Magnitude |
|---|---|---|
| Y-Intercept | 7.0000 | |
| Slope at Origin | 3.0000 |
Function Types Reference
| Type | General Form | Y-Intercept | Max X-Intercepts |
|---|---|---|---|
| Linear | y = mx + b | b | 1 |
| Quadratic | y = ax² + bx + c | c | 2 |
| Cubic | y = ax³ + bx² + cx + d | d | 3 |
| Exponential | y = a·e^(bx) + c | a + c | 0 or 1 |
| Logarithmic | y = a·ln(x) + b | Undefined | 1 |
How the Y-Intercept Is Found
| Step | Description |
|---|---|
| 1 | Set x = 0 in the function equation. |
| 2 | Simplify all terms containing x (they become 0). |
| 3 | The remaining value is the y-intercept (the constant term). |
| 4 | The y-intercept point is (0, f(0)). |
| 5 | For logarithmic functions, f(0) is undefined because ln(0) = −∞. |
Intercept Rules Quick Reference
| Rule | Explanation |
|---|---|
| Every polynomial has a y-intercept | Constant term = f(0) |
| Degree n → up to n x-intercepts | A polynomial of degree n has at most n real roots |
| Exponential y = a·e^(bx) | Y-intercept = a; never zero unless shifted |
| Logarithmic y = ln(x) | No y-intercept; x-intercept at x = 1 |
| Rational functions | Y-intercept exists if x = 0 is in the domain |
Planning notes, formulas, and examples
About the Y-Intercept Calculator
The y-intercept of a function is the point where its graph crosses the y-axis, which occurs when x = 0. Finding intercepts is one of the most fundamental skills in algebra and analytic geometry because intercepts anchor the graph to the coordinate plane and provide concrete reference points for sketching curves. For a polynomial like y = ax² + bx + c, the y-intercept is simply the constant term c, while for exponential functions like y = a·e^(bx) + c the y-intercept is a + c. Logarithmic functions are a special case: because ln(0) is undefined, they have no y-intercept at all.
This calculator supports five common function families — linear, quadratic, cubic, exponential, and logarithmic — and computes not only the y-intercept but also x-intercepts, the slope (derivative) at the origin, domain, and range. Visual comparison bars help you compare intercept size across function types, and the reference table summarizes the rule for each family. It is useful when you want to check how coefficients move a graph, confirm an intercept before graphing by hand, or compare how different function types behave at x = 0.
When This Page Helps
Y-intercepts are often the first anchor point you plot when sketching a graph, but the surrounding context matters too. This calculator keeps the y-intercept next to the x-intercepts, slope at the origin, domain, and range so you can check whether a function behaves the way you expect before graphing or differentiating further.
It is especially useful when you want to compare function families. A quadratic, cubic, exponential, and logarithmic equation can all be entered with the same goal in mind: see what happens at x = 0, confirm whether a y-intercept exists at all, and connect that result to the rest of the graph.
How to Use the Inputs
- Enter Slope (m) and Constant (b) in the input fields.
- Select the mode, method, or precision options that match your y-intercept problem.
- Read Y-Intercept first, then use X-Intercept(s) to confirm your setup is correct.
- Try a preset such as "Linear (y = mx + b)" to test a known case quickly.
Formula used
Y-intercept = f(0)
Linear: b
Quadratic: c
Cubic: d
Exponential (a·e^(bx)+c): a + c
Logarithmic (a·ln(x)+b): Undefined (ln(0) = −∞)
Slope at origin = f′(0)Example Calculation
Result: Y-Intercept shown by the calculator
Using the preset "Linear (y = mx + b)", the calculator evaluates the y-intercept setup, applies the selected algebra rules, and reports Y-Intercept with supporting checks so you can verify each transformation.
Tips & Best Practices
- For any polynomial, the y-intercept is always the constant term — the coefficient with no x attached.
- Exponential functions always cross the y-axis (unless the base function is shifted to exactly cancel out).
- Logarithmic functions never have a y-intercept because ln(0) is undefined; they do have an x-intercept.
- The number of x-intercepts of a polynomial cannot exceed its degree.
- When graphing, plot the y-intercept first — it is the easiest point to find.
How This Y-Intercept Calculator Works
The calculator evaluates each function at x = 0, which is the defining step for finding a y-intercept. For polynomials, that means reading the constant term. For exponential functions, it means substituting x = 0 into the exponential expression. For logarithmic functions, it checks domain restrictions and reports that no y-intercept exists because the function is undefined at x = 0.
Interpreting Results
Start with the y-intercept, then compare it with the x-intercepts and the slope at the origin. That combination tells you where the graph crosses the axes, whether it is increasing or decreasing at the y-axis, and whether the intercept size makes sense for the coefficients you entered.
Study Strategy
Use the presets to compare how different equation families behave at x = 0. Then change one coefficient at a time and watch which outputs move. That is an efficient way to see how constants, growth terms, and leading coefficients affect intercepts and local behavior.
Sources & Methodology
Last updated:
Frequently Asked Questions
The y-intercept is the point where a function's graph crosses the y-axis. It occurs at x = 0, so you find it by evaluating f(0). The y-intercept is written as the ordered pair (0, f(0)).
For y = ax² + bx + c, substitute x = 0: y = a(0)² + b(0) + c = c. The y-intercept is simply the constant term c, giving the point (0, c).
Because ln(0) is undefined (it approaches negative infinity). The natural log function is only defined for x > 0, so its graph never reaches the y-axis.
The y-intercept is where the graph crosses the y-axis (x = 0). The x-intercept is where the graph crosses the x-axis (y = 0). A function can have at most one y-intercept but may have multiple x-intercepts.
No. By the vertical line test, a function assigns exactly one output to each input. Since the y-axis is the vertical line x = 0, a function can cross it at most once.
In slope-intercept form y = mx + b, the constant b is the y-intercept. This is the most common reason this form is taught — you can read the y-intercept directly from the equation.
The slope at the origin is f′(0), the derivative evaluated at x = 0. It tells you the instantaneous rate of change of the function at the y-intercept — whether the graph is rising, falling, or flat at that point.
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