How to Graph a Quadratic Function: Steps and Explanation

graph of a quadratic function

A quadratic function is a polynomial function of degree 2

A quadratic function is a polynomial function of degree 2. It can be written in the general form:

f(x) = ax^2 + bx + c

Here, ‘a’, ‘b’, and ‘c’ are constants, and ‘x’ is the variable. The graph of a quadratic function is called a parabola, and its shape depends on the values of ‘a’, ‘b’, and ‘c’.

To graph a quadratic function, there are a few key steps to follow:

Step 1: Determine the vertex of the parabola. The vertex represents the highest or lowest point on the graph. The formula to find the vertex is given by:

x = -b / (2a) (for the x-coordinate of the vertex)
y = f(x) (to find the y-coordinate of the vertex, substitute the x-coordinate into the function)

Step 2: Find the y-intercept. The y-intercept is the point where the parabola crosses the y-axis, and it can be found by substituting x = 0 into the function:

y-intercept = f(0) = c

Step 3: Determine the x-intercepts (if any). The x-intercepts are the points where the parabola crosses the x-axis. They can be found by solving the quadratic equation ax^2 + bx + c = 0. This can be done by factoring, completing the square, or using the quadratic formula.

Step 4: Plot the vertex, y-intercept, and x-intercepts (if any) on a coordinate plane. These points help us determine the shape and position of the parabola.

Step 5: Determine additional points to plot. You can choose any x-values and find the corresponding y-values by substituting them into the quadratic function. It’s generally helpful to choose x-values symmetrically around the vertex to create a smoother graph.

Step 6: Connect the points to form the parabolic curve. Since a quadratic function is continuous, you can smoothly connect the points to form the graph of the parabola.

Remember, the values of ‘a’, ‘b’, and ‘c’ in the quadratic function can affect the orientation, width, and position of the parabola. Positive ‘a’ values make the parabola open upwards (concave up), while negative ‘a’ values make it open downwards (concave down). The values of ‘a’ and ‘b’ also affect the steepness and direction of the curve.

By following these steps, you can graph any quadratic function accurately.

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