What Is Vertex Form? Unlocking the Elements of Quadratic Functions!
Quadratic functions are essential building blocks of algebra. There are different forms of quadratic functions with each form exhibiting unique properties. The vertex form is one of those forms and through the vertex form, we can see key insights about the quadratic function and its graph. In this article, let’s discover the different elements of quadratic functions through its vertex form.
What Is the Vertex Form of the Quadratic Expression?
The vertex form of the quadratic expression or a quadratic function is rewritten to highlight the coordinates of the vertex. When talking about quadratic functions, we often think of its standard form, y = ax2+bx + c. Now, it’s time to put a spotlight on its vertex form: y = a(x -h)2+k, where (h, k) represents the vertex of the parabola and a still represents the leading coefficient of the quadratic expression.
Remember that we call the U-shaped curve the parabola, which represents a quadratic function’s graph. Now, the minimum (or maximum) point of the graph is what we call the vertex. The coordinate of the vertex, as we have shown, plays an important role in setting up the equation for the quadratic function’s vertex form.
Through the quadratic function’s vertex form, it’s easy for us to determine the coordinates of the vertex and whether it’s opening upward or downward. That makes graphing the quadratic function easier too. Don’t worry, we’ll break down these components for you to help you appreciate this form more!
Understanding the Components of the Vertex Form
There are a lot of key insights about the quadratic function from its vertex form. This is why it’s essential that we understand what these components represent. Recall that the vertex form of a quadratic expression (or function) is y = a(x -h)2+k. It’s now time to focus on what we know about the function given the two components: a and (h, k).
When Does the Parabola Open Upward or Downward?
The leading coefficient, a, tells you whether the parabola opens upward or downward. Observe the sign of the leading coefficient then use that to determine whether the quadratic function’s graph opens upward or downward.
If a is positive (or a > 0), the parabola is expected to open upward. This also means that the parabola’s vertex represents the function’s minimum value.
If a is negative (or a < 0), the parabola opens downward, so the parabola’s vertex represents the function’s maximum value.
By inspecting the sign of a, it’s easy for us to predict whether the quadratic function’s vertex represents its minimum or maximum point. Now, it’s time to see how the values of h and k help with the translation of the y=x2’s graph.
How Does the Vertex Form Affect the Graph of y=x2?
The vertex form immediately highlights the vertex, (h, k), of the quadratic expression. This also tells us the translations done one the parent function, y=x2, both vertically and horizontally. Given the vertex form of the quadratic function, y=a(x - h)2 + k, we can infer the following:
If h>0, the graph of y=x2, shifts h units to the right. Meanwhile, when h<0, the graph of y=x2, shifts h units to the left.
If k>0, the graph of y=x2, shifts k units upward. Meanwhile, when k<0, the graph of y=x2, shifts k units downward.
This shows that by using the vertex form of the quadratic function, it’s easy for us to have a rough idea of how the function’s graph would look like. For example, f(x)= -(x-3)2 -1, is in its vertex form. By inspecting the components of the quadratic function, we have the following information:
a=-1, so the parabola opens downward
(h, k) = (3, -1), so the graph of f(x) is simply the result of translating the graph of y=x2 3 units to the right and 1 unit downward.
This example shows one of the many benefits of knowing the vertex form of a quadratic function. You can also reverse the process and determine the equation of the quadratic function given its graph. Apply what you know to find the equation representing the graph shown below.
First, find the vertex of the parabola. For this graph, we can that it’s opening upward and have its lowest point at (2, 2). This means that the vertex is (2, 2). Use the vertex form of the quadratic function, y=a(x -h)2+k.
y= a(x - 2)2 + 2
Now, to find the value of a, use another point that the graph passes through like (0, 4). Substitute (0, 4) into the vertex form of the quadratic equation.
0= a(4 - 2)2 + 2
0= 4a + 2
4a = -2
a = -12
Substitute a = -12 in to the vertex form. Hence, we have y=-12(x - 2)2 + 2. This means that given the graph of the quadratic expression, we can now find the function’s quadratic expression.
How Do You Convert a Quadratic Function From Its Standard Form to Its Vertex Form?
There are different ways to rewrite a quadratic function in standard form to its vertex form. One way is by completing the square and another would be to use the vertex formula. Don’t worry, we’ll cover both in this article.
When given a quadratic function in standard form, y=ax2+bx+c, we can rewrite it in its vertex form, y=a(x-h)2+k. While the first method (completing the square) is more tedious, it’s essential as it establishes how we came up with the vertex formula.
Writing the Vertex Form Using Completing the Square
When given a quadratic function in standard form, y=ax2+bx+c, we can rewrite it in its vertex form, y=a(x-h)2+k. While the first method (completing the square) is more tedious, it’s essential as it establishes how we came up with the vertex formula. We’ll break down the steps for you by rewriting y=2x2-6x+12 in its vertex form.
- If the leading coefficient, a, is not 1, factor it out from the terms except the constant.
- Now that you have a binomial within the parenthesis, complete the square by dividing the second term by 2 and then squaring the result.
- Whatever constant you add, “balance” the expression by subtracting the same value outside the parenthesis. Don’t forget to include the factor outside.
- The trinomial inside the parenthesis is now a perfect square trinomial. Apply the algebraic property, (a b)2=a22ab+b2, to factor the expression.
This leads us to the vertex form of the quadratic equation.This shows that by completing the square, it’s possible for us to write any quadratic equation in standard form to its vertex form. In fact, use this method to come up with the vertex formula!
it’s essential as it establishes how we came up with the vertex formula.
Writing the Vertex Form Using the Vertex Formula
We can also find the vertex form of the quadratic expression, y=2x2-6x+12, using the vertex formula. The vertex formula uses the coefficients of the quadratic function in standard form, so it’s a convenient way to find its vertex.
Given a quadratic function in standard form, y=ax2+ bx + c, use the coefficients to find its vertex. Since a remains the same for both forms, it will now straightforward to rewrite it in its vertex form, y=a(x -h)2+ k. Let’s try this for y=2x2-6x+12!
- Find the x-coordinate of the vertex by using the vertex formula, h = -b2a .
- Now, find the y-coordinate of the vertex by applying the vertex formula, k= -D4a , or evaluating the function at x=32.
- Use the coordinates of the vertex and the value of the leading coefficient to write the vertex form of y=2x2-6x+12.
This shows that regardless of the method, we’ll end up with the same vertex form of the quadratic expression. Knowing both methods gives you the edge!
Take It Further With Juni Learning
We’ve now given you all the building blocks that you need to master this topic. Understand the essence of the vertex form and remember the ways it can help you in navigating more complex topics in Algebra. Take one step further and check Juni Learning’s comprehensive materials on quadratic functions and more complicated math topics. Expand your knowledge, master the fundamentals, and take on challenging topics through their math courses.