Chapter 3: Q 52. (page 156)

Determine the quadratic function whose graph is given.

Short Answer

Expert verified

The quadratic function is $f (x) = - 2 x^{2} - 8 x - 2$ .

Step by step solution

Step 1. Given information.

The given graph is:

Step 2. Determine the vertex from the given graph.

By analyzing the graph of the function, the vertex of the function is $(- 2, 6)$ .

So, $(h, k) = (- 2, 6)$ .

The quadratic form of the function is:

$\begin{array}{rcl} f (x) & = & a {(x - h)}^{2} + k \\ f (x) & = & a {(x - (- 2))}^{2} + 6 \\ f (x) & = & a {(x + 2)}^{2} + 6 \dots \dots \dots \dots (h = - 2, k = 6) \end{array}$

Step 3. Determine the value of a using the additional point 0,-2.

Substitute $x = 0$ and $f (0) = - 2$ in the above function.

$\begin{array}{rcl} f (x) & = & a {(x + 2)}^{2} + 6 \\ f (0) & = & a {(0 + 2)}^{2} + 6 \\ - 2 & = & 4 a + 6 \\ 4 a + 6 & = & - 2 \\ 4 a & = & - 2 - 6 \\ 4 a & = & - 8 \\ a & = & \frac{- 8}{4} \\ a & = & - 2 \end{array}$

Step 4. Now, substitute a=-2 in fx=ax+22+6.

$\begin{array}{rcl} f (x) & = & a {(x + 2)}^{2} + 6 \\ = & - 2 {(x + 2)}^{2} + 6 \\ = & - 2 (x^{2} + 4 x + 4) + 6 \\ = & - 2 x^{2} - 8 x - 8 + 6 \\ = & - 2 x^{2} - 8 x - 2 \end{array}$

Step 5. Simplified answer.

Hence, the required quadratic function is $\begin{array}{rcl} f (x) & = & - 2 x^{2} - 8 x - 2 \end{array}$ .

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Determine the quadratic function whose graph is given.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the vertex from the given graph.

Step 3. Determine the value of a using the additional point 0,-2.

Step 4. Now, substitute a=-2 in fx=ax+22+6.

Step 5. Simplified answer.

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