How to Sketch the Graph of a Quadratic Function: Step-by-Step Guide

graph of a quadratic function

A quadratic function is a polynomial of degree 2, expressed as f(x) = ax^2 + bx + c, where a, b, and c are constants

A quadratic function is a polynomial of degree 2, expressed as f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is called a parabola.

To sketch the graph of a quadratic function, you can follow these steps:

Step 1: Determine the vertex:
The vertex of a quadratic function is given by the coordinates (h, k), where h = -b/2a and k = f(h). The vertex represents the highest or lowest point on the parabola.

Step 2: Find the y-intercept:
The y-intercept is the point where the parabola intersects the y-axis. To find it, let x = 0 and solve for y. The y-intercept is then given by the coordinate (0, c).

Step 3: Determine the x-intercepts (if any):
The x-intercepts are the points where the parabola intersects the x-axis. To find them, let y = 0 and solve the quadratic equation ax^2 + bx + c = 0 using factoring, completing the square, or the quadratic formula.

Step 4: Plot additional points:
You can choose other x-values and substitute them into the equation to get the corresponding y-values to plot more points on the graph. This will help you create a more accurate representation of the parabola.

Step 5: Sketch the parabola:
Using the information obtained from the previous steps, plot the vertex, y-intercept, x-intercepts, and any additional points you calculated. Then, connect the dots smoothly to form a U-shape or an upside-down U-shape, depending on the sign of the coefficient of x^2.

Additionally, the coefficient a determines the “steepness” or “width” of the parabola:
– If a > 0, the parabola opens upward and is narrower.
– If a < 0, the parabola opens downward and is wider. Remember to label the axes and provide an appropriate scale for your graph. It's important to mention that while these steps explain the general process of graphing a quadratic function, sometimes certain transformations (like translations, stretches/compressions, or reflections) might be applied to the original function, which will affect the resulting graph.

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