Problem 97 - Algebra and Trigonometry

Question

Write the equation of each parabola in standard form. Vertex: \((-3,-4) ;\) The graph passes through the point \((1,4)\)

Step-by-Step Solution

Answer

The equation of the parabola in standard form is \(y = \frac{2}{5}(x + 3)^2 - 4\).

Step 1: Input the vertex into the equation

First, we need to insert the vertex into the standard equation. Our vertex is (-3,-4), so h is -3 and k is -4. Therefore, after inserting, our equation is \(y = a(x - (-3))^2 - 4\), which simplifies to \(y = a(x + 3)^2 - 4\).

Step 2: Find 'a' by using the given point

The given point that the graph passes through is (1,4). We will substitute these values into the equation to solve for 'a'. Inserting x=1 and y=4, we get: \(4 = a(1 + 3)^2 - 4\). Simplifying and solving the equation gives: \( a= \frac{2}{5}\).

Step 3: Write the final equation in standard form

Now that we have a value for 'a' as \(\frac{2}{5}\), We substitute this value back to the equation we got in step 1 to get the final equation of the parabola: \(y = \frac{2}{5}(x + 3)^2 - 4\).