Solving Quadratic Equations
The standard form of a quadratic equation is y equals ax squared plus bx plus c. a, b and c are the constants. When the constants are changed the graph changes. With a the coefficient of the x squared term, when a is positive the graph is concave up.
That’s this first graph and the last graph they both have positive coefficients in this case 1, a is 1 for both of these graphs and 1 is positive. Both of these parabolas are concave up. When a is negative the parabola is concave down like we have here in the second graph, y equals negative 2x squared, concave down.
This first graph is the graph of the parent function. It’s when none of the constants have been changed a is one, there is no b constant or c they’re both 0. This is the parent function before any shifts have happened. The larger the a is, the greater the absolute value of a, the steeper the graph is, the faster it increases or decreases.
In the first and the last graph it’s 1, in this middle graph the a is -2, and you can tell it’s skinnier than these two graphs are because 2 is larger than 1, it’s increasing or actually in this case decreasing more rapidly.
If a was say 1/2 instead of 2, then the graph would be a wider graph or a shallower graph. It wouldn’t be increasing or decreasing as rapidly. The c constant shifts the graph vertically, either up or down.
In both the first and the second graph the c is zero, and our graph passes through the origin and both those actually that’s where the vertex is. However, in this last graph notice there is no x term. In this last graph, our c is 2 and our graph has been shifted up to vertically 2.
The c will always be the y intercept of the quadratic equation, but it won’t always be where the vertex is. If there is an x term in there then that’s not going to be where your vertex is. The only reason the vertex is the same as the c is because there is no x term.
That’s the same for all of these, the c is zero for both of these graphs. Notice the vertex is at zero but that’s because neither one of these have an x term either.
When you throw that middle term in there, that x term, then it starts moving the graphs around. However, the c will always be the y intercept, it just won’t always be the vertex as well as the y intercept.