f(x) graph is in upward u shape in quadrant 1 with a vertex of (3,1) and points (2,2), (4,2), (1,4), and (3,4)

Question

g(x) graph is an upward u shape in quadrant 4,1 and slightly in 2 with vertex (3,-6) and points (1,-2), (5,-2), ((0,3) and ((6,3)
If g(x)=f(x)+j find the value of k

Answer 1

Given the properties of the functions \(f(x)\) and \(g(x)\), we can derive some aspects of their relationships.

Given Information:

1. \(f(x)\) has its vertex at \((3, 1)\) and its graph opens upward. It's in the first quadrant and has points at:

   • \((2, 2)\)
   • \((4, 2)\)
   • \((1, 4)\)
   • \((3, 4)\)

2. \(g(x)\) has its vertex at \((3, -6)\) and opens upward as well. It exists in quadrants 4, 1, and slightly in 2, with points at:

   • \((1, -2)\)
   • \((5, -2)\)
   • \((0, 3)\)
   • \((6, 3)\)

Finding Relationship Between Functions:

From the information, we know that:

\[ g(x) = f(x) + j \]

where \(j\) is a constant. To find \(j\), we can look at the vertices of both functions since they share the same \(x\)-coordinate (3) due to how they are defined.

For \(f(3)\):

• At \(x = 3\), \(f(3) = 1\) (the \(y\)-coordinate of the vertex).

For \(g(3)\):

• At \(x = 3\), \(g(3) = -6\) (the \(y\)-coordinate of the vertex).

Setting Up the Equation:

Since \(g(x) = f(x) + j\):

\[ g(3) = f(3) + j \]

Substituting the known values:

\[ -6 = 1 + j \]

Solving for \(j\):

To find \(j\):

\[ j = -6 - 1 = -7 \]

Conclusion:

Thus, the relationship we derived indicates that:

\[ g(x) = f(x) - 7 \]

This means the value of \(j\) is \( -7 \).

Answer:

The value of \(j\) is \( -7\).