Diana G. answered • 03/03/20
Experienced High school math teacher
Assuming that you mean y = x^2 + 4x + 4...
There are many parts to the graph of a parabola. Let's start with the x-intercepts (or zeros) -- these occur when the function is equal to 0 so we need to solve the equation x^2 + 4x + 4 = 0. This equation is easily factorable (x+2)(x+2)=0, and using the zero-product property this can only be true when one of the factors is 0. Since both factors are the same, that means x+2=0, so x=-2 and the parabola touches the x-axis only at (-2,0).
Axis of symmetry is a vertical line whose equation is described by the formula x = -b / 2a. In our case, a=1 and b=4. Applying the formula we have x = -(4) / (2·1) or x=-2.
Now the vertex is on the line of symmetry, so its x-coordinate is -2. We can find its y-coordinate (just as for any other point on the parabola) by plugging into the function rule: y = (-2)^2 + 4(-2) + 4 or y=0. So the vertex of this parabola is the point (-2,0). Does this look familiar? The vertex is the same point where the parabola touches the x-axis!!! [Note: that is not normal, it just happens to be true in this case]
What about the y-intercept? Since this point is on the y-axis, we know that its x-coordinate must be 0. So plug in 0 to find the y-coordinate: y = (0)^2 + 4(0) + 4 or y=4. Did you notice that all of the terms had no value except for the constant? This will always happen whenever you plug in x=0. The y-intercept is (0,4).
But wait, there's more! Since a parabola is always symmetrical, we can find another point that is a reflection of the y-intercept across the line of symmetry. The line of symmetry was at x=-2 and the y-intercept is at x=0 (two spaces to the right of the line). That means the mirror image of the point will be two spaces to the left of the line, or x=-4. It will be the same number of spaces above the x-axis as the y-intercept was, so its coordinates are (-4,4).
Altogether, this gives us 3 points to connect together, making a "U-shape" as expected for a quadratic function. Additionally, the leading coefficient (a) is a positive number (1) which indicates that the parabola opens upward. This matches the idea suggested by the locations of the 3 points we plotted.
