4. Polynomial Functions

Quadratic Functions

4. Polynomial Functions

# Quadratic Functions - Video Tutorials & Practice Problems

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concept

## Properties of Parabolas

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Problem

ProblemIdentify the ordered pair of the vertex of the parabola. State whether it is a minimum or maximum.

A

$\left(-3,-2\right);$ maximum

B

$\left(-3,-2\right);$ minimum

C

$\left(-1,2\right);$ maximum

D

$\left(-1,2\right);$ minimum

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Problem

ProblemWhere is the axis of symmetry located on the given parabola?

A

$x=3$

B

$x=0$

C

$x=1$

D

$x=4$

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concept

## Vertex Form

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5

example

## Example 1

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6

Problem

ProblemGraph the given quadratic function. Identify the vertex, axis of symmetry, intercepts, domain, range, and intervals for which the function is increasing or decreasing. $f\left(x\right)=-\left(x-5\right)^2+1$

A

B

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D

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concept

## Converting Standard Form to Vertex Form

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8

example

## Example 2

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9

Problem

ProblemGraph the given quadratic function. Identify the vertex, axis of symmetry, intercepts, domain, range, and intervals for which the function is increasing or decreasing. $f\left(x\right)=3x^2+12x$

A

B

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D

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PRACTICE PROBLEMS AND ACTIVITIES (67)

- In Exercises 1–4, the graph of a quadratic function is given. Write the function's equation, selecting from th...
- In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equat...
- In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equat...
- In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equat...
- Graph each quadratic function. Give the vertex, axis, x-intercepts, y-intercept, domain, range, and largest op...
- In Exercises 1–4, the graph of a quadratic function is given. Write the function's equation, selecting from th...
- Fill in the blank(s) to correctly complete each sentence. The highest point on the graph of a parabola that op...
- Fill in the blank(s) to correctly complete each sentence. The vertex of the graph of ƒ(x) = x^2 + 2x + 4 has x...
- In Exercises 5–8, the graph of a quadratic function is given. Write the function's equation, selecting from th...
- In Exercises 5–6, use the function's equation, and not its graph, to find (a) the minimum or maximum value and...
- In Exercises 5–8, the graph of a quadratic function is given. Write the function's equation, selecting from th...
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each mon...
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each mon...
- Solve each problem. During the course of ayear, the number of volunteers available to run a food bank each mon...
- In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function...
- Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is ...
- In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function...
- Consider the graph of each quadratic function.(a) Give the domain and range.
- Consider the graph of each quadratic function.(a) Give the domain and range.
- In Exercises 9–16, find the coordinates of the vertex for the parabola defined by the given quadratic function...
- Match each function with its graph without actually entering it into a calculator. Then, after completing the ...
- Match each function with its graph without actually entering it into a calculator. Then, after completing the ...
- In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equ...
- In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equ...
- In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equ...
- Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x...
- Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x...
- Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x...
- In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the ...
- In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the ...
- In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the ...
- In Exercises 45–48, give the domain and the range of each quadratic function whose graph is described. The ver...
- Several graphs of the quadratic function ƒ(x) = ax^2 + bx + c are shown below. For the given restrictions on a...
- In Exercises 45–48, give the domain and the range of each quadratic function whose graph is described. Maximum...
- Several graphs of the quadratic function ƒ(x) = ax^2 + bx + c are shown below. For the given restrictions on a...
- In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f...
- Several graphs of the quadratic function ƒ(x) = ax^2 + bx + c are shown below. For the given restrictions on a...
- In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f...
- Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note follo...
- In Exercises 53–56, write an equation in vertex form of the parabola that has the same shape as the graph of f...
- Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note follo...
- In Exercises 53–56, write an equation in vertex form of the parabola that has the same shape as the graph of f...
- Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the max...
- Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is ...
- Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity...
- Find a value of c so that y = x^2 - 10x + c has exactly one x-intercept.
- For what values of a does y = ax^2 - 8x + 4 have no x-intercepts?
- Define the quadratic function ƒ having x-intercepts (2, 0) and (5, 0) and y-intercept (0, 5).
- Define the quadratic function ƒ having x-intercepts (1, 0) and (-2, 0) and y-intercept (0, 4).
- The distance between the two points P(x₁, y₁) and R(x₂, y₂) is d(P, R) = √(x₁ - x₂)^2 + (y₁ -y₂)^2. Distance f...
- A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). What is the other solution o...
- In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-4) The graph passes throu...
- In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes throu...