Question

1. chelsea transformed the function ( f(x) = -x^2 - 14x - 52 ) into ( f(x) = -(x + 7)^2 - 3 ). she then wrote four statements about the function. which statement is incorrect?

• the axis of symmetry is ( x = -7 ).
• the minimum value of the function is ( -7 ).
• the vertex is located at ( (-7, -3) ).
• the maximum value of the function is ( -3 ).

Explanation:

Step1: Recall the vertex form of a quadratic function

The vertex form of a quadratic function is \(f(x)=a(x - h)^2 + k\), where \((h,k)\) is the vertex of the parabola, and the axis of symmetry is \(x = h\). If \(a>0\), the parabola opens upward and has a minimum value of \(k\); if \(a<0\), the parabola opens downward and has a maximum value of \(k\).

Step2: Analyze the given function \(f(x)=-(x + 7)^2-3\)

• For the axis of symmetry: Comparing with the vertex form \(f(x)=a(x - h)^2 + k\), we have \(h=-7\), so the axis of symmetry is \(x=-7\). This statement "The axis of symmetry is \(x = - 7\)" is correct.
• For the vertex: From the vertex form \(f(x)=-(x + 7)^2-3=a(x - h)^2 + k\), we get \(h=-7\) and \(k = - 3\), so the vertex is \((-7,-3)\). The statement "The vertex is located at \((-7,-3)\)" is correct.
• For the maximum or minimum value: Since \(a=-1<0\), the parabola opens downward, so the function has a maximum value. The maximum value is \(k=-3\), not a minimum value of \(-7\). So the statement "The minimum value of the function is \(-7\)" is incorrect. The statement "The maximum value of the function is \(-3\)" is correct.

Answer:

The minimum value of the function is \(-7\) (this statement is incorrect).