How to Identify Vertex Form of Quadratic Functions
Quick Answer
To identify the vertex form of a quadratic function, look for the format y = a(x - h)² + k, where (h, k) is the vertex. Among given options, only the one in this form directly reveals the vertex without rewriting.
When dealing with quadratic functions, it’s important to recognize their forms, especially when you're asked to identify features like the vertex. The vertex of a quadratic function is a critical point that determines the function's maximum or minimum value. The vertex form of a quadratic function is expressed as y = a(x - h)² + k, where (h, k) represents the vertex coordinates.
Let's break down the options given in the original question to see which one reveals the vertex without needing any manipulation.
1. **Factored Form**: The first choice, g(x) = -2(x - 6)(x - 2), is in factored form. This form is useful for finding the x-intercepts (the points where the graph crosses the x-axis), but it does not directly show the vertex. To find the vertex from this form, you would have to expand it and complete the square, which is not efficient for immediate identification of the vertex.
2. **Partially Factored/Standard Mix**: The second option, g(x) = -2(x² - 8x + 12), is a mix but does not clearly indicate the vertex. This expression is close to standard form but would require further manipulation to determine the vertex.
3. **Standard Form**: The third option, g(x) = -2x² + 16x - 24, is in standard form (ax² + bx + c). While you can calculate the vertex from standard form using the formula h = -b/(2a), it requires additional steps that do not allow for immediate recognition of the vertex.
4. **Vertex Form**: Lastly, the fourth option, g(x) = -2(x - 4)² + 8, is in vertex form. Here, you can see directly that the vertex is at the point (4, 8). This makes it the best choice when you're asked to select the form that reveals the vertex without having to rewrite it.
Understanding how to identify the vertex form is crucial for solving quadratic equations and analyzing their graphs. Quadratics appear in various real-world situations, such as projectile motion, where the vertex represents the maximum height reached by an object.
In summary, when tasked with identifying the vertex of a quadratic function, always look for the vertex form. It allows you to quickly find the vertex and understand the graph's behavior without unnecessary calculations. Practice recognizing these forms, and you will become more adept at working with quadratic functions in your math studies.
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