Chapter 3: Q 74. (page 157)

The graph of the function $f (x) = a x^{2} + b x + c$ has vertex at $(1, 4)$ and passes through the point $(- 1, - 8)$ . Findrole="math" localid="1646894617982" $a, b$ and $c$ .

Short Answer

Expert verified

$a = - 3, b = 6, c = 1$

Step by step solution

Step 1. Given information.

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

Given the vertex $(1, 4)$ and a point $(- 1, - 8)$ .

Step 2. Determine the value of a.

The vertex of a parabola is $(h, k) = (1, 4)$ . So, $h = 1$ and $k = 4$ .

The vertex form of the parabola is:

$f (x) = a {(x - h)}^{2} + k$

We know that one additional point $(- 1, - 8)$ .

So, $x = - 1$ and $\begin{array}{rcl} f (x) & = & f (- 1) \\ = & - 8 \end{array}$

The required value of $a$ is:

$\begin{array}{rcl} f (x) & = & a {(x - h)}^{2} + k \\ f (- 1) & = & a {(- 1 - 1)}^{2} + 4 \dots \dots \dots (x = - 1, h = 1, k = 4, f (- 1) = - 8) \\ - 8 & = & a {(- 2)}^{2} + 4 \\ - 8 & = & 4 a + 4 \\ 4 a & = & - 12 \\ a & = & \frac{- 12}{4} \\ a & = & - 3 \end{array}$

Step 3. Determine the value of b and c.

$\begin{array}{rcl} f (x) & = & a {(x - h)}^{2} + k \\ f (x) & = & - 3 {(x - 1)}^{2} + 4 \dots \dots (h = 1, k = 4, a = - 3) \\ f (x) & = & - 3 (x^{2} - 2 x + 1) + 4 \\ f (x) & = & - 3 x^{2} + 6 x - 3 + 4 \\ f (x) & = & - 3 x^{2} + 6 x + 1 \end{array}$

Compare the function $f (x) = - 3 x^{2} + 6 x + 1$ with $f (x) = a x^{2} + b x + c$ .

Hence, $a = - 3, b = 6$ and $c = 1$ .

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The graph of the function $f (x) = a x^{2} + b x + c$ has vertex at $(1, 4)$ and passes through the point $(- 1, - 8)$ . Findrole="math" localid="1646894617982" $a, b$ and $c$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the value of a.

Step 3. Determine the value of b and c.

One App. One Place for Learning.

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The graph of the function fx=ax2+bx+chas vertex at 1,4and passes through the point -1,-8. Findrole="math" localid="1646894617982" a,bandc.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determine the value of a.

Step 3. Determine the value of b and c.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

The graph of the function $f (x) = a x^{2} + b x + c$ has vertex at $(1, 4)$ and passes through the point $(- 1, - 8)$ . Findrole="math" localid="1646894617982" $a, b$ and $c$ .