Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 39

Chapter 7, Problem 39

Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.

Verified step by step guidance

Start by writing the general form of the quadratic function: $f(x) = ax^{2} + bx + c$.

Use the given points to create a system of equations by substituting each $x$ and $f(x)$ value into the quadratic function:

For $f(-2) = -4$, substitute $x = -2$ and $f(x) = -4$ to get: $a(-2)^{2} + b(-2) + c = -4$.

For $f(1) = 2$, substitute $x = 1$ and $f(x) = 2$ to get: $a(1)^{2} + b(1) + c = 2$.

For $f(2) = 0$, substitute $x = 2$ and $f(x) = 0$ to get: $a(2)^{2} + b(2) + c = 0$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Its graph is a parabola, and the coefficients determine its shape and position.

Function Evaluation

Function evaluation involves substituting a specific input value into the function to find the corresponding output. For example, f(−2) means replacing x with −2 in the quadratic expression and calculating the result.

Solving Systems of Equations

To find the coefficients a, b, and c, you set up equations based on the given function values and solve the resulting system of linear equations. This process often uses substitution or elimination methods to find the unknowns.

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Textbook Question

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases}\)x - y + z = 8 \\2y - z = -7 \\2x + 3y = 1$\end{cases}\]\text{The inverse of }\[\begin{bmatrix}$1 & -1 & 1 \\0 & 2 & -1 \\2 & 3 & 0$\end{bmatrix}\]\text{ is }\[\begin{bmatrix}$3 & 3 & -1 \\-2 & -2 & 1 \\-4 & -5 & 2\(\end{bmatrix}\]\text{.}$

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Textbook Question

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}\)4x - 5y - 6z = -1 $\x$ - 2y - 5z = -12 \\2x - y = 7\(\end{cases}$

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Textbook Question

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

Textbook Question

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$

BC + CB

Textbook Question

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

Textbook Question

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases}\)w - x + 2y $\quad\]\quad$ = -3 $\quad\[\quad$ x - y + z = 4 \\-w + x - y + 2z = 2 $\quad\]\quad$ -x + y - 2z = -4$\end{cases}\[\text{The inverse of }\]\begin{bmatrix}$1 & -1 & 2 & 0 \\0 & 1 & -1 & 1 \\-1 & 1 & -1 & 2 \\0 & -1 & 1 & -2$\end{bmatrix}\[\text{ is }\]\begin{bmatrix}$0 & 0 & -1 & -1 \\1 & 4 & 1 & 3 \\1 & 2 & 1 & 2 \\0 & -1 & 0 & -1\(\end{bmatrix}$