Questions: Determine the quadratic function of the form f(x)=a(x-h)^2+k whose graph is given on the right. f(x)=

Transcript text: Determine the quadratic function of the form $f(x)=a(x-h)^{2}+k$ whose graph is given on the right. \[ f(x)= \]

Solution

Solution Steps

To determine the quadratic function in the form $ f(x) = a(x-h)^2 + k $, we need to identify the vertex $(h, k)$ and a point on the graph to solve for $a$. The vertex form of a quadratic function is useful because it directly gives us the vertex, and we can use another point to find the value of $a$.

Step 1: Identify the Vertex

From the given information, we have the vertex of the quadratic function as $ (h, k) = (2, 3) $.

Step 2: Use a Point to Find $ a $

We are given a point on the graph, $ (x_1, y_1) = (4, 7) $. We can use this point to find the value of $ a $ in the vertex form of the quadratic function $ f(x) = a(x - h)^2 + k $.

Substituting the values into the equation: \[ y_1 = a(x_1 - h)^2 + k \] \[ 7 = a(4 - 2)^2 + 3 \] This simplifies to: \[ 7 = a(2)^2 + 3 \] \[ 7 = 4a + 3 \] Subtracting 3 from both sides gives: \[ 4 = 4a \] Dividing by 4 results in: \[ a = 1 \]

Step 3: Write the Quadratic Function

Now that we have $ a $, $ h $, and $ k $, we can write the complete quadratic function: \[ f(x) = 1(x - 2)^2 + 3 \] This simplifies to: \[ f(x) = (x - 2)^2 + 3 \]