The directions are : Write each equation in standard form (if needed), then find appropriate information for the particular conic.
the question: y=2x^2-12x+19
HOW DO I BEGIN???
Complete the square and rewrite as
y = 2x^2-12x + 19
= 2(x^2 - 6x + 9)+ 19 - 18
= 2(x-3)^2 + 1
This is an upward-pointing parabola with its lowest point at x=3, y+1
If you define new coordinates as
y'= y-1 and x' = x-3,
the equation becomes
y' = 2 x'^2
To begin, you can use the process of completing the square to rewrite the equation in standard form.
1. Start by isolating the terms with x. In this case, you already have y on one side, so you can leave it as it is: y = 2x^2 - 12x + 19.
2. Focus on the terms containing x. To complete the square, you need to factor out the coefficient of x^2 from the first two terms. In this case, it is 2. Write it outside of the parentheses: y = 2(x^2 - 6x) + 19.
3. Now, you need to add and subtract a specific value inside the parentheses to complete the square. Take half of the coefficient of x in the original equation (which is -6), square it, and add it inside the parentheses. This value is (-6/2)^2 = 9: y = 2(x^2 - 6x + 9 - 9) + 19.
4. The next step is to simplify the equation. The terms inside the parentheses can be factored as a perfect square trinomial: y = 2((x - 3)^2 - 9) + 19.
5. Distribute the 2 to both terms inside the parentheses: y = 2(x - 3)^2 - 18 + 19.
6. Combine the constants -18 and 19: y = 2(x - 3)^2 + 1.
Now, the equation is in standard form. It represents an upward-pointing parabola with its vertex at the point (3, 1).
To find appropriate information for the particular conic, you can determine the type of conic the equation represents. In this case, the equation is a quadratic function, indicating a parabola. The coefficient of x^2 (2) tells you that the parabola is vertically stretched. The positive sign indicates that it opens upward.
If you need specific information about the parabola, such as the focus, directrix, or axis of symmetry, you would need additional information or another form of the equation (such as vertex form or general form) to find those details.