Given the graph of a quadratic​ function, write the equation of the function in vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.

Question

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Part 1
Given the graph of a quadratic​ function, write the equation of the function in vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.
x
y
left parenthesis 0 comma 0 right parenthesisleft parenthesis 1 comma negative 6 right parenthesis

A coordinate system has a horizontal x-axis and a vertical y-axis. The graph is a parabola with vertex (1,negative 6) passing through the origin.
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Part 1
​f(x)equals
  
enter your response here ​(Simplify your​ answer.)
y= (0,0)
x= (1,-6)

Answer 1

To write the equation of the quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), we first need to identify the vertex \((h, k)\) and the value of \( a \).

From the information provided, we know that the vertex of the parabola is at the point \((1, -6)\). Therefore, \( h = 1 \) and \( k = -6 \).

Now, the equation looks like this: \[ f(x) = a(x - 1)^2 - 6 \]

Next, we can use another point on the graph to find the value of \( a \). The point \((0, 0)\) lies on the graph, which means \( f(0) = 0 \).

Substituting \( x = 0 \) and \( f(0) = 0 \) into the equation gives us: \[ 0 = a(0 - 1)^2 - 6 \] \[ 0 = a(1) - 6 \] \[ a - 6 = 0 \] \[ a = 6 \]

Now we can substitute \( a \) back into our equation: \[ f(x) = 6(x - 1)^2 - 6 \]

So, the equation of the quadratic function in vertex form is: \[ f(x) = 6(x - 1)^2 - 6 \]