4. Polynomial Functions

Define the quadratic function ƒ having x-intercepts (1, 0) and (-2, 0) and y-intercept (0, 4).

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2(x+2)²−1

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)²+5

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f\left(x\right)=-x^2+2x+3$

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.

$f\left(x\right)=(x+1{)}^2-1g\left(x\right)=(x+1{)}^2+1$

$h\left(x\right)=(x-1{)}^2+1j\left(x\right)=(x-1{)}^2-1$

Consider the graph of each quadratic function.

(a) Give the domain and range.

Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x² but with the given point as the vertex. (5, 3)

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=4−(x−1)²

Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by $s\left(t\right)=-16t^2+64t+100$. Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

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

Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V\left(x\right)$, where $V\left(x\right)=2x^2-32x+150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V\left(x\right)$ is modeled by $V\left(x\right)=31x-226$. Find the number of volunteers in each of the following months. Sketch a graph of $y=V\left(x\right)$ for January through December. In what month are the fewest volunteers available?

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.

$f\left(x\right)=x^2+2x+1g\left(x\right)=x^2-2x+1$

$h\left(x\right)=x^2-1j\left(x\right)=-x^2-1$

Give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.