Online Function Grapher

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graph of a quadratic function

Graph of a quadratic function: parabola, vertex and roots

Type the function in the tool, for example x^2 - 4x + 3, and the parabola is drawn with its opening direction, vertex, roots and the point where it crosses the y axis. To see what each coefficient does, type a*x^2 + b*x + c and move the sliders for a, b and c.

What defines the graph of ax² + bx + c

The graph of f(x) = ax² + bx + c, with a not equal to zero, is a parabola. The sign of a sets the opening direction: with a positive the parabola opens upward and the vertex is the lowest point; with a negative it opens downward and the vertex is the highest point. The larger the absolute value of a, the narrower the curve: y = 4x² is narrower than y = x², and y = 0.25x² is wider. The value of c is the y intercept, because f(0) = c, which is why x² - 4x + 3 crosses the y axis at 3.
The vertex sits at x = -b/(2a) and y = -Δ/(4a), where Δ = b² - 4ac is the discriminant. For x² - 4x + 3 we have a = 1, b = -4 and c = 3, so Δ = 16 - 12 = 4, xv = 4/2 = 2 and yv = -4/4 = -1. The vertical line x = 2 is the axis of symmetry: the points x = 1 and x = 3 have the same height, and you can see it on the graph. If a is positive the range runs from yv up to plus infinity; if it is negative it runs from minus infinity up to yv.
The roots are the solutions of f(x) = 0 by the quadratic formula: x = (-b ± √Δ)/(2a). With Δ greater than zero there are two real roots and the parabola crosses the x axis at two points. With Δ equal to zero there is a repeated root and the vertex touches the x axis, as in x² - 6x + 9 = (x - 3)², which has vertex (3, 0). With Δ less than zero there is no real root and the parabola stays entirely on one side of the axis, as in x² + 1, whose vertex is (0, 1). The tool shows Δ and writes the roots in exact form when the coefficients are integers or short decimals, for example (1 ± √5)/2 for x² - x - 1.
To compare two curves, add a second function. The parabola x² - 4x + 3 and the line x/2 + 1 cross where x² - 4x + 3 = x/2 + 1, that is x² - 4.5x + 2 = 0, which gives x = 0.5 and x = 4, with y = 1.25 and y = 3. These points show up as circles on the graph and in the list of intersections. The intervals come ready too: for this parabola the function is decreasing for x less than 2, increasing for x greater than 2 and negative only between the roots 1 and 3.

Checked examples

Parabola with two roots

Input: x^2 - 4x + 3
Expected output: x = 1, x = 3; V = (2, -1); f(0) = 3

Roots 1 and 3; vertex (2, -1); y = 3 at x = 0. Δ = 4, opens upward.

Opening downward

Input: -x^2 + 2x + 3
Expected output: x = -1, x = 3; V = (1, 4)

Roots -1 and 3; vertex (1, 4). a = -1, so the vertex is the highest point.

No real roots

Input: x^2 + 1
Expected output: Δ = -4; V = (0, 1)

No real roots; vertex (0, 1). Δ = -4, the parabola stays above the x axis.

Repeated root

Input: x^2 - 6x + 9
Expected output: x = 3; V = (3, 0); Δ = 0

Root 3; vertex (3, 0). Δ = 0, the parabola only touches the x axis.

Irrational roots

Input: x^2 - x - 1
Expected output: x = (1 ± √5)/2 ≈ -0.618, 1.618; V = (0.5, -1.25)

Roots (1 ± √5)/2, about -0.618 and 1.618; vertex (0.5, -1.25). Δ = 5, which is not a perfect square.

Parabola and line

Input: x^2 - 4x + 3 e x/2 + 1
Expected output: (0.5, 1.25), (4, 3)

Intersections (0.5, 1.25) and (4, 3). Solves x² - 4.5x + 2 = 0.

Full tool FAQ

How do I graph a quadratic function?

Type the function, for example x^2 - 4x + 3. The parabola opens upward if a is positive and downward if it is negative. The vertex is at x = -b/2a: here x = 2 and y = -1. The roots come from (-b ± √Δ)/2a with Δ = b² - 4ac = 4, so x = 1 and x = 3, and the parabola crosses the y axis at c = 3. The tool marks all of these points and shows their exact form.

How do I find the roots of a function on the graph?

How do I identify the asymptotes of a function?

What is the difference between an asymptote and a hole in a graph?

How do I plot several graphs and find where they cross?

What are the parameters a, b and c?

Which functions and symbols can I use?

Does the panel show the important points even off screen?

How do I save or share the graph?

Why does my function not show up or give an error?

Frequently asked questions

How can I tell if the parabola opens up or down?

Look at the sign of the coefficient a, the number multiplying x². If a is positive (x² - 4x + 3, a = 1) the parabola opens upward and has a minimum point. If a is negative (-x² + 2x + 3, a = -1) it opens downward and has a maximum point. The tool states the opening direction and says whether the vertex is a minimum or a maximum.

How do I find the vertex without drawing the graph?

Compute xv = -b/(2a) and then plug it into the function. In f(x) = 2x² - 4x - 6, xv = 4/4 = 1 and yv = 2 - 4 - 6 = -8, so the vertex is (1, -8). The tool does this automatically and also shows the axis of symmetry, x = 1 in this case.

What does a negative Δ mean on the graph?

It means the parabola does not cross the x axis: there is no real root. In x² + 1 we have Δ = 0 - 4 = -4 and the graph stays entirely above the axis, with the vertex at (0, 1). The roots exist in the complex plane (±i) but do not show up on the real graph.

Next steps

Properties

f₁(x) = x^2 - 4x + 3Quadratic

Quadratic function, f(x) = ax² + bx + c with a not equal to 0. The graph is a parabola, symmetric about the vertical line through the vertex. Here a = 1, b = -4 and c = 3; the parabola opens upward.

Exact result (algebra)

Domain: ℝ
Range: [-1, +∞)
Roots (zeros): x = 1
x = 3
y intercept: (0, 3)
Vertex: (2, -1) minimum point
Axis of symmetry: x = 2
Discriminant Δ: 4 (two real roots)
Increasing and decreasing: Decreasing on (-∞, 2]
Increasing on [2, +∞)
Sign: f₁(x) > 0 on (-∞, 1) ∪ (3, +∞)
f₁(x) < 0 on (1, 3)
Asymptotes: None found
Parity: Neither even nor odd.

f₂(x) = x/2 + 1Linear (1st degree)

Linear function, f(x) = ax + b. The graph is a line: a is the slope (increasing if a is positive) and b is where the line crosses the y axis. Here a = 0.5 (increasing) and b = 1.

Exact result (algebra)

Domain: ℝ
Range: ℝ
Roots (zeros): x = -2
y intercept: (0, 1)
Increasing and decreasing: Increasing on (-∞, +∞)
Sign: f₂(x) > 0 on (-2, +∞)
f₂(x) < 0 on (-∞, -2)
Asymptotes: None found
Parity: Neither even nor odd.

Intersections between the functions

f₁ ∩ f₂ (0.5, 1.25)
f₁ ∩ f₂ (4, 3)

x	f₁(x)	f₂(x)
-5	48	-1.5
-4	35	-1
-3	24	-0.5
-2	15	0
-1	8	0.5
0	3	1
1	0	1.5
2	-1	2
3	0	2.5
4	3	3
5	8	3.5

f₁(x)

f₂(x)

-5

-1.5

-4

-1

-3

-0.5

-2

-1

0.5

1.5

-1

2.5

3.5

What defines the graph of ax² + bx + c

The graph of f(x) = ax² + bx + c, with a not equal to zero, is a parabola. The sign of a sets the opening direction: with a positive the parabola opens upward and the vertex is the lowest point; with a negative it opens downward and the vertex is the highest point. The larger the absolute value of a, the narrower the curve: y = 4x² is narrower than y = x², and y = 0.25x² is wider. The value of c is the y intercept, because f(0) = c, which is why x² - 4x + 3 crosses the y axis at 3.

The vertex sits at x = -b/(2a) and y = -Δ/(4a), where Δ = b² - 4ac is the discriminant. For x² - 4x + 3 we have a = 1, b = -4 and c = 3, so Δ = 16 - 12 = 4, xv = 4/2 = 2 and yv = -4/4 = -1. The vertical line x = 2 is the axis of symmetry: the points x = 1 and x = 3 have the same height, and you can see it on the graph. If a is positive the range runs from yv up to plus infinity; if it is negative it runs from minus infinity up to yv.

The roots are the solutions of f(x) = 0 by the quadratic formula: x = (-b ± √Δ)/(2a). With Δ greater than zero there are two real roots and the parabola crosses the x axis at two points. With Δ equal to zero there is a repeated root and the vertex touches the x axis, as in x² - 6x + 9 = (x - 3)², which has vertex (3, 0). With Δ less than zero there is no real root and the parabola stays entirely on one side of the axis, as in x² + 1, whose vertex is (0, 1). The tool shows Δ and writes the roots in exact form when the coefficients are integers or short decimals, for example (1 ± √5)/2 for x² - x - 1.

To compare two curves, add a second function. The parabola x² - 4x + 3 and the line x/2 + 1 cross where x² - 4x + 3 = x/2 + 1, that is x² - 4.5x + 2 = 0, which gives x = 0.5 and x = 4, with y = 1.25 and y = 3. These points show up as circles on the graph and in the list of intersections. The intervals come ready too: for this parabola the function is decreasing for x less than 2, increasing for x greater than 2 and negative only between the roots 1 and 3.

Checked examples

Parabola with two roots

Input: x^2 - 4x + 3
Expected output: x = 1, x = 3; V = (2, -1); f(0) = 3

Roots 1 and 3; vertex (2, -1); y = 3 at x = 0. Δ = 4, opens upward.

Opening downward

Input: -x^2 + 2x + 3
Expected output: x = -1, x = 3; V = (1, 4)

Roots -1 and 3; vertex (1, 4). a = -1, so the vertex is the highest point.

No real roots

Input: x^2 + 1
Expected output: Δ = -4; V = (0, 1)

No real roots; vertex (0, 1). Δ = -4, the parabola stays above the x axis.

Repeated root

Input: x^2 - 6x + 9
Expected output: x = 3; V = (3, 0); Δ = 0

Root 3; vertex (3, 0). Δ = 0, the parabola only touches the x axis.

Irrational roots

Input: x^2 - x - 1
Expected output: x = (1 ± √5)/2 ≈ -0.618, 1.618; V = (0.5, -1.25)

Roots (1 ± √5)/2, about -0.618 and 1.618; vertex (0.5, -1.25). Δ = 5, which is not a perfect square.

Parabola and line

Input: x^2 - 4x + 3 e x/2 + 1
Expected output: (0.5, 1.25), (4, 3)

Intersections (0.5, 1.25) and (4, 3). Solves x² - 4.5x + 2 = 0.

Full tool FAQ

How do I graph a quadratic function?

How do I find the roots of a function on the graph?

How do I identify the asymptotes of a function?

What is the difference between an asymptote and a hole in a graph?

How do I plot several graphs and find where they cross?

What are the parameters a, b and c?

Which functions and symbols can I use?

Does the panel show the important points even off screen?

How do I save or share the graph?

Why does my function not show up or give an error?

Frequently asked questions