Key Features
In GQF Target B, we learned that there are three different forms of quadratic functions in which you can graph from. Those three forms are: standard form, intercept/factored form, and vertex form. Each of these forms can provide you with key features of the graph of a quadratic function. When a function is written in standard form, the \(y\)-intercept can be easily identified by looking at the \(c\) in the equation. When a function is written in intercept/factored form, the \(x\)-intercepts can easily identifiable, hence the name of the form: intercept form. And finally, when a function is written in vertex form, the vertex can easily be identified. This is why it is called vertex form!
In GQF Target B, we learned that there are three different forms of quadratic functions in which you can graph from. Those three forms are: standard form, intercept/factored form, and vertex form. Each of these forms can provide you with key features of the graph of a quadratic function. When a function is written in standard form, the \(y\)-intercept can be easily identified by looking at the \(c\) in the equation. When a function is written in intercept/factored form, the \(x\)-intercepts can easily identifiable, hence the name of the form: intercept form. And finally, when a function is written in vertex form, the vertex can easily be identified. This is why it is called vertex form!
Form |
Function |
Key Feature |
Standard Form |
\(f(x) = ax^2 + bx + c\) |
\(y-\)intercept: \((0, c)\) |
Intercept/Factored Form |
\(f(x) = a(x- p)(x - q)\) |
\(x-\)intercepts: \((p, 0)\) and \((q, 0)\) |
Vertex Form |
\(f(x) = a(x - h)^2 + k\) |
vertex: \((h, k)\) |
Look at the functions below and determine which form each function is written in, and the key feature that can be identified.
Function A: \(f(x) = (x – 5)(x + 2)\)
Function B: \(f(x) = 2(x – 3)^2 + 4\)
Function C: \(f(x) = x^2 + 3x + 10\)
Function A is written in intercept/factored form, therefore the x-intercepts can be found. The intercepts are \((5, 0)\) and \((-2, 0)\). Function B is written in vertex form, which means the key feature for this function would be the vertex \((3, 4)\). Function C is written in standard form, and the \(y-\)intercept can be found by looking at the “\(c\)” value: \((0, 10)\).
Now that we know the key features that can be found from each of the three different forms, let’s take a look at how to convert between each of the three forms. Why would this be beneficial to know? Let’s say you have a function written in standard form, for example, \(f(x) = x^2 – 6x + 8\). We know that the y-intercept of the graph of this function is located at \((0, 8)\) because the key feature for a function written in standard form is the y-intercept. But what if you were asked for the vertex of \(f(x) = x^2 – 6x + 8\)? You would need to rewrite the function in vertex form! In order to write equivalent quadratic function, use the guidelines below:
To convert from:
Factored form to Standard form: Distribute (Click here to review how to distribute)
Standard form to Factored form: Factor (Click here to review how to factor)
Standard form to Vertex form: Complete the Square (Click here to review how to complete the square)
Vertex form to Standard form: Expand & Combine Like Terms
Function A: \(f(x) = (x – 5)(x + 2)\)
Function B: \(f(x) = 2(x – 3)^2 + 4\)
Function C: \(f(x) = x^2 + 3x + 10\)
Function A is written in intercept/factored form, therefore the x-intercepts can be found. The intercepts are \((5, 0)\) and \((-2, 0)\). Function B is written in vertex form, which means the key feature for this function would be the vertex \((3, 4)\). Function C is written in standard form, and the \(y-\)intercept can be found by looking at the “\(c\)” value: \((0, 10)\).
Now that we know the key features that can be found from each of the three different forms, let’s take a look at how to convert between each of the three forms. Why would this be beneficial to know? Let’s say you have a function written in standard form, for example, \(f(x) = x^2 – 6x + 8\). We know that the y-intercept of the graph of this function is located at \((0, 8)\) because the key feature for a function written in standard form is the y-intercept. But what if you were asked for the vertex of \(f(x) = x^2 – 6x + 8\)? You would need to rewrite the function in vertex form! In order to write equivalent quadratic function, use the guidelines below:
To convert from:
Factored form to Standard form: Distribute (Click here to review how to distribute)
Standard form to Factored form: Factor (Click here to review how to factor)
Standard form to Vertex form: Complete the Square (Click here to review how to complete the square)
Vertex form to Standard form: Expand & Combine Like Terms
Example 1: Convert \(f(x) = (x + 7)(x - 3)\) into standard form.
Solution:
Distribute: \(f(x) = (x + 7)(x - 3) = x^2 + 4x - 21\)
Standard Form: \(f(x) = ax^2 +bx + c\)
The key feature for standard from is the \(y-\)intercept \((0, c)\) For this function the \(y-\)intercept of the graph is located at \((0, -21)\).
Solution:
Distribute: \(f(x) = (x + 7)(x - 3) = x^2 + 4x - 21\)
Standard Form: \(f(x) = ax^2 +bx + c\)
The key feature for standard from is the \(y-\)intercept \((0, c)\) For this function the \(y-\)intercept of the graph is located at \((0, -21)\).
Example 2: Convert \(f(x) = 2x^2 + 14x - 60\) into factored/intercept form.
Solution:
Factor a GCF: \(f(x) = 2x^2 + 14x - 60 = 2(x^2 + 7x - 30\)
Factor: \(f(x) = 2(x - 3)(x + 10)\)
Intercept/Factored Form: \(f(x) = a(x - p)(x - q)\)
The key feature for intercept form is the \(x-\)intercepts \((p, 0)\) and \(q, 0)\). For this function the \(x-\)intercepts are \(3, 0\) and \(-10, 0\).
Solution:
Factor a GCF: \(f(x) = 2x^2 + 14x - 60 = 2(x^2 + 7x - 30\)
Factor: \(f(x) = 2(x - 3)(x + 10)\)
Intercept/Factored Form: \(f(x) = a(x - p)(x - q)\)
The key feature for intercept form is the \(x-\)intercepts \((p, 0)\) and \(q, 0)\). For this function the \(x-\)intercepts are \(3, 0\) and \(-10, 0\).
Example 3: Convert \(f(x) = x^2 -4x + 7\) into vertex form.
Solution:
Isolate the quadratic and linear terms: \(f(x) - 7 = x^2 - 4x\)
Complete the square: \(f(x) - 7 + 4 = x^2 - 4x + 4\)
Write as a perfect square: \(f(x) - 3 = (x - 2)^2\)
Write in vertex form: \(f(x) = (x - 2)^2 + 3\)
Vertex form: \(f(x) = a(x - h)^2 + k\)
The key feature for vertex form is the vertex \((h, k)\). For this function the vertex of the graph is located at \((2, 3)\).
Solution:
Isolate the quadratic and linear terms: \(f(x) - 7 = x^2 - 4x\)
Complete the square: \(f(x) - 7 + 4 = x^2 - 4x + 4\)
Write as a perfect square: \(f(x) - 3 = (x - 2)^2\)
Write in vertex form: \(f(x) = (x - 2)^2 + 3\)
Vertex form: \(f(x) = a(x - h)^2 + k\)
The key feature for vertex form is the vertex \((h, k)\). For this function the vertex of the graph is located at \((2, 3)\).
Example 4: Convert \(f(x) = 2(x + 3)^2 - 5\) into standard form.
Solution:
Expand: \(f(x) = 2(x + 3)^2 - 5 = 2(x + 3)(x + 3) - 5\)
Distribute: \(f(x) = 2(x^2 + 6x + 9) - 5\)
Distribute the 2: \(f(x) = 2x^2 + 12x + 18 - 5\)
Combine like terms: \(f(x) = 2x^2 +12x + 13\)
Standard form: \(f(x) = ax^2 + bx + c\)
The key feature for standard form is the \(y-\)intercept \((0, c)\). For this function the \(y-\)intercept of the graph is located at \((0, 13)\).
Solution:
Expand: \(f(x) = 2(x + 3)^2 - 5 = 2(x + 3)(x + 3) - 5\)
Distribute: \(f(x) = 2(x^2 + 6x + 9) - 5\)
Distribute the 2: \(f(x) = 2x^2 + 12x + 18 - 5\)
Combine like terms: \(f(x) = 2x^2 +12x + 13\)
Standard form: \(f(x) = ax^2 + bx + c\)
The key feature for standard form is the \(y-\)intercept \((0, c)\). For this function the \(y-\)intercept of the graph is located at \((0, 13)\).
Quick Check
Identify the form that each function is in, and the key feature that can be found from that function.
1) \(f(x) = -3(x – 7)(x – 1)\)
2) \(g(x) = (x + 6)^2 – 2\)
Determine which form is needed. Then, write the equivalent function to find the key feature.
3) Find the vertex of \(k(x) = (x + 4)(x + 2)\)
4) Find the x-intercepts of \(h(x) = 2x^2 – 10x – 12\)
Quick Check Solutions
Identify the form that each function is in, and the key feature that can be found from that function.
1) \(f(x) = -3(x – 7)(x – 1)\)
2) \(g(x) = (x + 6)^2 – 2\)
Determine which form is needed. Then, write the equivalent function to find the key feature.
3) Find the vertex of \(k(x) = (x + 4)(x + 2)\)
4) Find the x-intercepts of \(h(x) = 2x^2 – 10x – 12\)
Quick Check Solutions