Week 9 – Solving a quadratic equation using a graph

This week in pre-calc, we learned how to take a quadratic equation, and be able to take the key points we need in order to graph the equation. There are 3 different equations that all give us different key points we need to graph. Vertex form ( y = a $(x-p)^2$+q ), general form ( y = a $x^2$ +bx + c ) and factored form ( y = a(x- $y_1$.)(x- $y_2$)

The vertex form gives you the vertex of the graph (the lowest or highest point of the graph). To find the vertex, all you do is take ‘p’ as your x value and because the ‘q’ is a constant term, that’s your y value. Vertex = (p,q)

The general form tells you the y-intercept and the direction the parabola is opening up. The y-intercept is always the constant number, and it’s always ‘c’. If there is no constant term (c) then the y-intercept is always 0. In the first term ‘ax’, the ‘a’ tells you the direction it’s opening towards. If the value is positive, it will be opening upwards. If the value is negative, it’ll open down. If it opens up, the vertex is the lowest point, and if it opens down, the vertex is the highest point. The value of ‘a’ can also tell you if the parabola is stretched or compressed. Stretched is any number greater than 1 and compressed is any number smaller than 1.

In factored form, you can find the x-intercepts. If you have y = 2(x – 3)(x + 2), because the numbers in the brackets can’t go lower, you can’t simplify anymore. You take your (x-3) and you find the value of ‘x’ that will make that term equal to 0. So 3-3 = 0, x-intercept is +3. You can just take the number (-3) and take its opposite sign to find the x-intercept too. Opposite of +2 is -2, x-intercept is -2. 