Question

Write the steps for solving a quadratic equation using a graph.

Short answer

The graph of a quadratic equation is a parabola that opens upwards if \(a>0\) and downwards if \(a<0\). The shape and position of the graph are determined by the vertex and the x-intercepts. The vertex \((-\frac{b}{2a}, f(-\frac{b}{2a})\) is the maximum or minimum point. The x-intercepts \((-b ± \sqrt{b^2 - 4ac})/(2a)\) are points where the graph touches the x-axis.

Step-by-step solution

Identify the Quadratic Equation

A quadratic equation is typically given in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\). Identify the equation that you will be graphing.

Find the Vertex

The vertex of a parabola is given by the point \((-\frac{b}{2a}, f(-\frac{b}{2a})\). The first coordinate of the vertex is given by the formula \(-\frac{b}{2a}\). Substitute the values of \(a\) and \(b\) from the equation to find the x-coordinate of the vertex. The second coordinate of the vertex is the value of the function at the x-coordinate. Simply substitute the x-coordinate into the equation to find the y-coordinate.

Find the x-Intercepts

The x-intercepts are solutions to the equation when \(y = 0\). In other words, solve \(ax^2 + bx + c = 0\), which is the Quadratic Formula \((-b ± \sqrt{b^2 - 4ac})/(2a)\). Substitute the values of \(a\), \(b\), and \(c\) into the formula to find the x-intercepts.

Draw the Graph

Now that you have the vertex and the x-intercepts, plot these points onto a graph. Draw a smooth curve through these points, making sure the curve is a symmetrical 'U' shape, or 'inverted U' shape if \(a\) is negative. The vertex should be the lowest or the highest point, and the graph should cut the x-axis at the x-intercepts.