Graphing quadratic functions is the process of plotting quadratic equations on a coordinate plane. A quadratic function has the standard form f of x equals a x squared plus b x plus c, where a, b, and c are constants and a is not equal to zero. The graph of any quadratic function is always a U-shaped curve called a parabola.
When graphing quadratic functions, we need to identify several key components. The vertex is the highest or lowest point of the parabola. The axis of symmetry is a vertical line that passes through the vertex. The y-intercept is where the graph crosses the y-axis, and x-intercepts are where it crosses the x-axis. The direction tells us whether the parabola opens upward when a is positive, or downward when a is negative.
To find the vertex of a parabola, we use the vertex formula. For a quadratic function f of x equals a x squared plus b x plus c, the x-coordinate of the vertex is negative b divided by 2a. Then we substitute this x-value back into the function to find the y-coordinate. For example, with f of x equals x squared minus 4x plus 3, we have a equals 1, b equals negative 4, and c equals 3. The vertex x-coordinate is negative negative 4 divided by 2 times 1, which equals 2. The y-coordinate is f of 2, which equals negative 1. So the vertex is at point 2, negative 1.
The parameter 'a' in a quadratic function controls both the direction and width of the parabola. When a is positive, the parabola opens upward, and when a is negative, it opens downward. The absolute value of a determines the width: larger values make the parabola narrower, while smaller values make it wider. Let's see how changing the value of a affects the shape of our parabola.
Let's put it all together with a complete example. To graph f of x equals x squared minus 2x minus 3, we follow these steps: First, find the vertex using the formula. The x-coordinate is 1, and the y-coordinate is negative 4. Second, find the y-intercept by setting x equal to zero, giving us the point 0, negative 3. Third, find the x-intercepts by solving the equation, which gives us negative 1, 0 and 3, 0. Finally, plot all these points and draw the parabola. This systematic approach ensures we capture all the key features of the quadratic function.