Graph in Standard Form: Quadratic Equations
Graph in Standard Form: Quadratic Equations - How it Works - Video
General Graph Information
General Graph Information:
Here we have the standard formula of a quadratic equation, y = ax2 + bx + c. Remember a and b are coefficients and c is a constant. The a value is the same as in the vertex form and intercept form. The great thing about c is that it tells us the y-intercept. Before we do any math, we know one point, (0, c).
In standard form, the next bit of information we can find is the axis of symmetry. In this case, with the parent function, the axis of symmetry is x = 0. The great thing about the axis of symmetry is that it tells us the x-part of the vertex. And, to find the y-part we substitute the value of the axis of symmetry into the equation to find the y-part, which is the maximum of minimum.
Maximum vs Minimum
Maximum vs Minimum:
To find out with the quadratic equation has a maximum or minimum point, we can look at the a-value. If a is positive, then the y-values go up, so the graph has a minimum. If the a is negative, then the y-values go down, so the graph has a maximum.
To the find the maximum/minimum value we look at the y-part of the vertex.
Example 1
Example 1:
Here we have the equation, y = x2 + 2x + 1.
The first step is write down the a, b, and c values. We can re-write the equation as y = 1x2 + 2x + 1 to help us identify the values. So a = 1, b = 2, and c = 1.
The great thing about the c-value is that it is the y-intercept so we already have one point, (0, 1).
Now we need to find the axis of symmetry so we can find the vertex. Remember the formula for the axis of symmetry is x = -b / (2a). So let's substitute 1 for a and 2 for b: -(2) / (2[1]), which is -1. So the axis of symmetry is x = -1.
Remember the axis of symmetry is the x-part of the vertex. Using that value, we can find y-part by substituting -1 for x into y = x2 + 2x + 1. After graphing axis of symmetry, we can graph the next point since the axis of symmetry is like a mirror. We can count to the axis of symmetry and in this case it is 1. So we count 1 away to find another point (-2, 1).
After substituting -1 for x, we get y = (-1)2 + 2(-1) + 1 => y = 1 - 2 + 1 => y = 0. With that value, we can create our vertex: (-1, 0).
Remember the vertex is either a maximum/minimum point. Since our a value is positive, we have a minimum value at y = 0.
Example 2
Example 2:
Here we have the equation, y = -x2 + 4x - 3.
The first step is write down the a, b, and c values. We can re-write the equation as y = -1x2 + 4x -3 to help us identify the values. So a = -1, b = 4, and c = -3.
The great thing about the c-value is that it is the y-intercept so we already have one point, (0, -3).
Now we need to find the axis of symmetry so we can find the vertex. Remember the formula for the axis of symmetry is x = -b / (2a). So let's substitute -1 for a and 4 for b: -(-4) / (2[1]), which is 2. So the axis of symmetry is x = 2.
Remember the axis of symmetry is the x-part of the vertex. Using that value, we can find y-part by substituting 2 for x into y = x2 + 2x + 1. After graphing axis of symmetry, we can graph the next point since the axis of symmetry is like a mirror. We can count to the axis of symmetry and in this case it is 1. So we count 2 away to find another point (4, -3).
After substituting 2 for x, we get y = -(2)2 + 4(2) - 3 => y = -4 + 8 - 3 => y = 1. With that value, we can create our vertex: (2, 1).
Remember the vertex is either a maximum/minimum point. Since our a value is negative, we have a maximum value at y = 1.