Concepts I touch on that you should be familiar with:

• Systems of equations
• Finding the y-intercept of a standard quadratic 
• Graphing functions

In algebra, school teaches you how to find the y-intercept of the function: plug in 0 for x and solve for y. Typically in a quadratic written in the form ax^2+bx+c, the y value of where the function crosses the y-axis is just c, because the other terms are multiplied by x and x=0. 

If that doesn’t stick in your head, it’s probably because no one has explained why it works. I love my visuals, so take a look at the function y = x^2 – 3x + 2 on a graph. 

By looking at it, it’s pretty clear the function crosses the y-axis at 2. This follows the pattern of the y value being the “c” in the given function. But we aren’t using the graph to see the y-intercept, because usually drawing the graph is too time consuming and there are many better ways to find that point. 

Remember how you were taught to plug in 0 for x? Well, what does that really mean? It means there are now two equations to consider. First is the original function (y = x^2 – 3x + 2) and the second is x = 0, which is what comes out of plugging 0 in for x. When given two different equations with at least one common variable, we can solve it like a system! 

Here we would use substitution because we have a value for x. We plug in 0 every time we see x in the original function, which gets rid of all the x values. Then we can just solve for y, and (0)^2 – 3(0) + 2 = 2. So our y value is 2. We were given our x value as 0 in the equation x = 0, so the y-intercept is (0,2). 

If that’s confusing, remember that solving a system of equations is just solving for the point in which the two equations intersect on a graph. So coming back to the original graph and graphing x = 0 on top of the original function we get:

The intersection point is at (0,2) which is also the y-intercept! 

Please comment or reach out with any questions!