We are going to graph a quadratic function from Standard Form. For the following function, let's look at the graph and see if we can find the key information that can be found directly from the equation.
Key Information
As you can see from the graph, the \(y\)-intercept is \((0, 6)\). If we look at the graph in standard form the \(y\)-intercept is the "\(c\)" value! The \(y\)-intercept is the key information that we can obtain directly from the equation. So without doing much work we can find the \(y\)-intercept.
After we find the \(y\)-intercept, we will need to find the axis of symmetry, vertex and then graph an additional point.
Let's try an example.
Example 1: Graph the following quadratic function: \(f(x) = x^2 - 8x + 12\). Identify the domain and range of the function.
Solution:
Key information
Recall the \(y\)-intercept is the \(c\) value so the \(y\)-intercept is \((0, 12)\).
Axis of Symmetry
In order to find the axis of symmetry we will use an equation that is derived from the standard form of a quadratic equation. The axis of symmetry is also the \(x\) value of the vertex which is \(x = \Large\frac{-b}{2a}\). In our function \(a = 1\), \(b = -8\). Let's substitute in these values to solve for the value of \(x\).
\(x = \Large\frac{-b}{2a}\)
\(x = \Large\frac{-(-8)}{2(1)} = \frac{8}{2}\)
\(x = 4\)
Vertex
We now have the \(x\) value of the vertex, \(x = 4\). In order to find the \(y\) value we will use substitution. Substitute the \(x\) value into the equation and solve for \(y\).
\(f(x) = x^2 - 6x +4\)
\(f(4) = (4)^2 - 8(4) + 12 = 16 - 32 + 12\)
\(f(4) = -4\)
So the vertex is \((4, -4)\)
Additional Point - Use Symmetry
To find one more point we will use symmetry. Once we graph the vertex, the line of symmetry and the \(y\)-intercept we can use symmetry to find the additional point. The point \((0, 12)\) is \(4\) units to the left of the axis of symmetry. So the additional point is \(4\) units to the right of the axis of symmetry with the same \(y\)-coordinate: \((4, 12)\). Now we can sketch the graph.
As you can see from the graph, the \(y\)-intercept is \((0, 6)\). If we look at the graph in standard form the \(y\)-intercept is the "\(c\)" value! The \(y\)-intercept is the key information that we can obtain directly from the equation. So without doing much work we can find the \(y\)-intercept.
After we find the \(y\)-intercept, we will need to find the axis of symmetry, vertex and then graph an additional point.
Let's try an example.
Example 1: Graph the following quadratic function: \(f(x) = x^2 - 8x + 12\). Identify the domain and range of the function.
Solution:
Key information
Recall the \(y\)-intercept is the \(c\) value so the \(y\)-intercept is \((0, 12)\).
Axis of Symmetry
In order to find the axis of symmetry we will use an equation that is derived from the standard form of a quadratic equation. The axis of symmetry is also the \(x\) value of the vertex which is \(x = \Large\frac{-b}{2a}\). In our function \(a = 1\), \(b = -8\). Let's substitute in these values to solve for the value of \(x\).
\(x = \Large\frac{-b}{2a}\)
\(x = \Large\frac{-(-8)}{2(1)} = \frac{8}{2}\)
\(x = 4\)
Vertex
We now have the \(x\) value of the vertex, \(x = 4\). In order to find the \(y\) value we will use substitution. Substitute the \(x\) value into the equation and solve for \(y\).
\(f(x) = x^2 - 6x +4\)
\(f(4) = (4)^2 - 8(4) + 12 = 16 - 32 + 12\)
\(f(4) = -4\)
So the vertex is \((4, -4)\)
Additional Point - Use Symmetry
To find one more point we will use symmetry. Once we graph the vertex, the line of symmetry and the \(y\)-intercept we can use symmetry to find the additional point. The point \((0, 12)\) is \(4\) units to the left of the axis of symmetry. So the additional point is \(4\) units to the right of the axis of symmetry with the same \(y\)-coordinate: \((4, 12)\). Now we can sketch the graph.
Domain: \(\mathbb{R}\) Range: \(y\geq -4\)
Example 2: Graph the quadratic function \(y = -2x^2 +12x - 7\)
Solution:
Watch the video for the solution.
Example 2: Graph the quadratic function \(y = -2x^2 +12x - 7\)
Solution:
Watch the video for the solution.