The given quadratic equation forms a parabola when graphed, a U-shaped line.
To find the minimum value of y, we could find the y-coordinate of the vertex.
<h3>What is the vertex?</h3>
The vertex represents the maximum or minimum point on a parabola. It tells us the optimal y-value of the function and the x-value at which it occurs.
For the given quadratic, the optimal y-value, or the y-coordinate of the vertex, will be a <em>minimum</em>. The value is positive, meaning the parabola will open upwards. (Please see below for an image representation).
<h2>Different Ways to Approach the Problem</h2>
There are multiple ways to approach this question.
One way is to rearrange the equation, which is currently in standard form, to vertex form.
- <em>Standard form for a quadratic</em>:
- <em>Vertex form for a quadratic</em>: where (h,k) is the vertex
Another way is to factor the equation to first find the x-intercepts of the graph, find the x-coordinate of the vertex, then at the end, plug the x-coordinate into the equation to find the minimum y-value.
The second strategy is a bit redundant for this type of question, so we can go about solving it by using the first method.
<h2>Finding the Vertex of this Parabola</h2>
First, we can group and by using a set of parentheses:
Then, we can complete the square:
Apply the rule :
Therefore, the vertex of the quadratic is (-3,-2).
This makes the minimum y-value -2. The x-value at which this occurs is -3.
<h2>Answer:</h2>
The minimum y-value -2. The x-value at which this occurs is -3.