Questions: Identify the vertex and leading coefficient of the quadratic function. Then write the expression as f(x)=a x^2+b x+c. f(x)=-3(x-5)^2+1 The vertex is (Type an ordered pair.)

Identify the vertex and leading coefficient of the quadratic function. Then write the expression as f(x)=a x^2+b x+c.
f(x)=-3(x-5)^2+1

The vertex is
(Type an ordered pair.)

Transcript text: Identify the vertex and leading coefficient of the quadratic function. Then write the expression as $f(x)=a x^{2}+b x+c$. \[ f(x)=-3(x-5)^{2}+1 \] The vertex is $\square$ (Type an ordered pair.)

Solution

Solution Steps

To identify the vertex and leading coefficient of the given quadratic function, we can use the standard form of a quadratic function $ f(x) = a(x-h)^2 + k $, where $(h, k)$ is the vertex and $a$ is the leading coefficient. Then, we can expand the expression to write it in the form $ f(x) = ax^2 + bx + c $.

Solution Approach

Identify the vertex $(h, k)$ from the given function.
Identify the leading coefficient $a$.
Expand the given function to write it in the form $ f(x) = ax^2 + bx + c $.

Step 1: Identify the Vertex

The given quadratic function is expressed in vertex form as $ f(x) = -3(x - 5)^2 + 1 $. From this form, we can identify the vertex as $ (h, k) = (5, 1) $.

Step 2: Determine the Leading Coefficient

The leading coefficient $ a $ of the quadratic function is the coefficient of the squared term. In this case, $ a = -3 $.

Step 3: Expand the Function

To express the function in standard form $ f(x) = ax^2 + bx + c $, we expand the given function: \[ f(x) = -3(x - 5)^2 + 1 = -3(x^2 - 10x + 25) + 1 = -3x^2 + 30x - 74 \]