Chapter 3: Q 73. (page 157)

The graph of the function $f (x) = a x^{2} + b x + c$ has vertex at $(0, 2)$ and passes through the point $(1, 8)$ . Find $a, b$ and $c$ .

Short Answer

Expert verified

The solutions are $a = 6, b = 0$ , and $c = 2$ .

Step by step solution

Step 1. Given information.

Given function $f (x) = a x^{2} + b x + c$ .

Vertex at $(0, 2)$ and passes through the point $(1, 8)$ .

Step 2. Determine the value of b and c.

The coordinates of the vertex of a quadratic function of the form $f (x) = a x^{2} + b x + c$ are role="math" localid="1646894449869" $(\frac{- b}{2 a}, f (\frac{- b}{2 a}))$ .

So, $(\frac{- b}{2 a}, f (\frac{- b}{2 a})) = (0, 2) \dots \dots (G i v e n v e r t e x (0, 2))$ .

$\begin{array}{rcl} \frac{- b}{2 a} & = & 0 \\ - b & = & 0 \\ b & = & 0 \end{array}$ and $\begin{array}{rcl} f (\frac{- b}{2 a}) & = & 2 \\ f (0) & = & 2 \dots \dots (\frac{- b}{2 a} = 0) \\ a {(0)}^{2} + b (0) + c & = & 2 \dots \dots (s u b s t i t u t e x = 0 i n f (x) = a x^{2} + b x + c) \\ c & = & 2 \end{array}$

Step 3. Determine the value of a.

From the point $(1, 8)$ , we find that $f (1) = 8$ .

Substitute $1$ for $x$ , $0$ for $b$ , $2$ for $c$ , and $8$ for $f (1)$ in $f (x) = a x^{2} + b x + c$ .

$\begin{array}{rcl} f (x) & = & a x^{2} + b x + c \\ f (1) & = & a {(1)}^{2} + 0 (1) + 2 \\ 8 & = & a + 2 \\ a + 2 & = & 8 \\ a & = & 8 - 2 \\ a & = & 6 \end{array}$

Step 4. Simplified answer.

Hence, $a = 6, b = 0$ , and $c = 2$ .

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The graph of the function $f (x) = a x^{2} + b x + c$ has vertex at $(0, 2)$ and passes through the point $(1, 8)$ . Find $a, b$ and $c$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the value of b and c.

Step 3. Determine the value of a.

Step 4. Simplified answer.

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The graph of the function fx=ax2+bx+chas vertex at 0,2and passes through the point 1,8. Finda,bandc.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the value of b and c.

Step 3. Determine the value of a.

Step 4. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

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The graph of the function $f (x) = a x^{2} + b x + c$ has vertex at $(0, 2)$ and passes through the point $(1, 8)$ . Find $a, b$ and $c$ .