Graph of Quadratic Function

A quadratic function is a function whose highest power of the variable is 2. It is written in the form f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0.

The graph of a quadratic function is called a parabola, a U-shaped curve that may open upward or downward.

Key Characteristics of Quadratic Functions

1. Direction: The direction of a parabola depends on the coefficient a. If a > 0, the parabola opens upward, while if a < 0, it opens downward.
2. Vertex: The vertex is the highest or lowest point on the parabola. It represents the maximum value for a downward-opening parabola and the minimum value for an upward-opening parabola.
3. Axis of Symmetry: The axis of symmetry is a vertical line passing through the vertex that divides the parabola into two equal and symmetrical halves.
4. x-intercepts: These are the points where the parabola crosses the x-axis. They represent the solutions or roots of the quadratic equation.
5. y-intercept: This is the point where the parabola crosses the y-axis and indicates the value of the function when x = 0.
6. Domain and Range: The domain of a quadratic function consists of all real numbers. The range depends on the position of the vertex and the direction in which the parabola opens.

Steps to Graph a Quadratic Function

1. Find the vertex of the parabola.
2. Determine the axis of symmetry.
3. Find the x-intercepts, if any.
4. Find the y-intercept.
5. Plot the points on the coordinate plane.
6. Draw a smooth parabola through the plotted points.

Graph of Quadratic Function Cases

The graph of a quadratic equation has two cases, which are as follows:

1. Upward Case (a > 0)

The direction of the graph completely depends upon the value of the coefficient of x², i.e., 'a'. If a is greater than zero, then the parabola thus formed will open upwards.

Example: Plot a graph of the quadratic equation y = 5x - 5.

Solution:

We have the equation: y = 5x² - 5, on comparing it with f(x) = ax² + bx + c

we have, a = 5, b = 0 and c = -5.

The vertex of the above equation is:
x = -b/(2a)
x = -0/(2(5))
x = 0

Now put x = 0 in the equation y = 5x² - 5
y = 5(0)² - 5
y = 0 - 5
y = -5

The vertex of the above equation is (0, -5).

Now, find the different values of x and y by solving the equation:

x	0	1	-1
y	-5	0	0

Plot the graph with these coordinates, the graph thus obtained will be a parabola opening upwards as a = 5 >0.

2. Downward Case (a < 0)

The direction of the parabola formed for the given quadratic equation will be oriented downwards if the value of the coefficient of x², i.e., 'a,' is less than zero.

Example: Plot a graph of quadratic equation y = -3(x + 2)² + 4.

Solution:

We have the equation: y = -3(x + 2)² + 4, on comparing it with a(x – h)² + k = 0

we have, h = -2, k = 4 and a = -3

The vertex of the above equation is given by (h, k), so vertex is (-2, 4)

Also a = -3, the negative value represent the downward direction of the parabola and so the vertex (-2, 4) is the point of absolute maxima.

Now, find the different values of x and y by solving the equation:

x	-3	-1	-2
y	1	1	4

Plot the graph with these coordinates, the graph thus obtained will be a parabola opening downwards as a = -3 < 0.

Graphing Quadratic Functions in Vertex Form

A quadratic function in vertex form is written as

f(x) = a(x − h)² + k

This form makes it easy to identify the vertex of the parabola, which is located at (h, k). To graph the function, plot the vertex and a few additional points, then draw a smooth curve through them. The resulting graph is a parabola that opens upward when a > 0 and downward when a < 0.

Steps to Graph a Quadratic Function in Vertex Form

1. Draw the x-axis and y-axis.
2. Identify and plot the vertex (h, k).
3. Find a few additional points by substituting values of x into the function.
4. Plot the points on the coordinate plane.
5. Draw a smooth parabola through the plotted points.

Solved Examples

Example 1: Draw the graph of the quadratic equation y = 3x + x.

Solution:

We have the equation: y = 3x² + x, on comparing it with f(x) = ax² + bx + c

we have, a = 3, b = 1 and c = 0.

The vertex of the above equation is:

x = -b/(2a)
x = -1/(2(3))
x = -1/ 6
x = -0.166

Now put x = -0.166 in the equation y = 3x² + x

y= 3(-0.166)² + (-0.166).
y = 3(0.0275) – 0.166
y = 0.0825 – 0.166
y = -0.0835

The vertex of the above equation is (-0.166, -0.0835)

Now, find the different values of x and y by solving the equation:

x	0	1	-1
y	0	4	2

Plot the graph with these coordinates, the graph thus obtained will be a parabola opening upwards as a = 3 >0.

Example 2: Determine the axis of symmetry and the y-intercept of the quadratic function f(x) = 5x² + 4x + 1.

Solution:

We have the equation: y = 5x² + 4x +1
Here, a = 5, b = 4 and c = 1

The axis of symmetry is given by x = -b/2a, putting values we get,
x = -4/2(5)
x = -4/10
x = -0.4

y-intercept is given by (0, c)

Here c = 1, so y-intercept = (0, 1).
Hence, the axis of symmetry and the y-intercept of the quadratic function f(x) = 5x² + 4x +1 is -0.4 and (0, 1) respectively.

Important Formulas

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