5.1 Quadratic Functions

$f \left(x\right) = 2 x^{2} - 4 x - 3$

Find the vertex of the quadratic function $f \left(x\right) = 2 x^{2} – 6 x + 7.$ Rewrite the quadratic in standard form (vertex form).

Given the equation $g \left(x\right) = 13 + x^{2} - 6 x ,$ write the equation in general form and then in standard form.

Find the domain and range of $f \left(x\right) = - 5 x^{2} + 9 x - 1.$

Find the domain and range of $f \left(x\right) = 2 \left(x - \frac{4}{7}\right)^{2} + \frac{8}{11} .$

The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the pr…

Find the y- and x-intercepts of the quadratic $f \left(x\right) = 3 x^{2} + 5 x - 2.$

In a Try It, we found the standard and general form for the function $g \left(x\right) = 13 + x^{2} - 6 x .$ Now find the y- and x-intercepts (if any).

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation $H \left(t\right) = −16 t^{2} + 96 t + 112.$

1. ⓐWhen does the rock reach the maximum height?
2. ⓑWhat is the maximum height of the rock?
3. ⓒWhen does the rock hit the ocean?

Explain the advantage of writing a quadratic function in standard form.

Explain why the condition of $a \neq 0$ is imposed in the definition of the quadratic function.

What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

Rewrite the quadratic functions in standard form and give the vertex.

$g \left(x\right) = x^{2} + 2 x - 3$

Rewrite the quadratic functions in standard form and give the vertex.

$f \left(x\right) = x^{2} + 5 x - 2$

Rewrite the quadratic functions in standard form and give the vertex.

$k \left(x\right) = 3 x^{2} - 6 x - 9$

Rewrite the quadratic functions in standard form and give the vertex.

$f \left(x\right) = 3 x^{2} - 5 x - 1$

Determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

$f \left(x\right) = 2 x^{2} - 10 x + 4$

Determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

$f \left(x\right) = 4 x^{2} + x - 1$

Determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

$f \left(x\right) = \frac{1}{2} x^{2} + 3 x + 1$

Determine the domain and range of the quadratic function.

$f \left(x\right) = \left(x - 3\right)^{2} + 2$

Determine the domain and range of the quadratic function.

$f \left(x\right) = x^{2} + 6 x + 4$

Determine the domain and range of the quadratic function.

$k \left(x\right) = 3 x^{2} - 6 x - 9$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(−2 , −1\right) , \left(x , y\right) = \left(−4 , 3\right)$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(2 , 3\right) , \left(x , y\right) = \left(5 , 12\right)$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(3 , 2\right) , \left(x , y\right) = \left(10 , 1\right)$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(1 , 0\right) , \left(x , y\right) = \left(0 , 1\right)$

Use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

Use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

Use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

Use a calculator to find the answer.

Graph on the same set of axes $f \left(x\right) = x^{2} , f \left(x\right) = \left(x - 2\right)^{2} , f \left(x - 3\right)^{2}$, and $f \left(x\right) = \left(x + 4\right)^{2} .$

What appears to be the effect of adding or subtracting those numbers?

Use a calculator to find the answer.

A suspension bridge can be modeled by the quadratic function $h \left(x\right) = .0001 x^{2}$ with $−2000 \leq x \leq 2000$ where $\left|x\right|$ is the number of feet from the center and $h \left(x\right)$ is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

Use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $\left(−1 , 2\right)$ opens down.

Use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $\left(−100 , 100\right) ,$ opens up.

Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\left(−1 , 4\right)$ and has the shape of $f \left(x\right) = 2 x^{2} .$ Vertex is on the $y -$ axis.

Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\left(1 , −3\right)$ and has the shape of $f \left(x\right) = - x^{2} .$ Vertex is on the $y -$ axis.

Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\left(1 , −6\right)$ has the shape of $f \left(x\right) = 3 x^{2} .$ Vertex has x-coordinate of $−1.$

Find the dimensions of the rectangular dog park split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

Suppose that the price per unit in dollars of a cell phone production is modeled by $p = \$ 45 - 0.0125 x ,$ where $x$ is in thousands of phones produced, and the revenue represented by thousands of dollars is $R = x \cdot p .$ Find the production level that will maximize revenue.

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by $h \left(t\right) = - 4.9 t^{2} + 24 t + 8.$ How long does it take to reach maximum height?

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Find the $x -$ intercepts of the quadratic function $f \left(x\right) = 2 x^{2} + 4 x - 4.$

Solve $x^{2} + x + 2 = 0.$

Solve the equations over the complex numbers.

$x^{2} = - 8$

Solve the equations over the complex numbers.

$x^{2} + 27 = 0$

Solve the equations over the complex numbers.

$x^{2} - 4 x + 5 = 0$

Solve the equations over the complex numbers.

$x^{2} - 4 x + 13 = 0$

Solve the equations over the complex numbers.

$x^{2} - 10 x + 26 = 0$

Solve the equations over the complex numbers.

$x \left(x - 4\right) = 20$

Solve the equations over the complex numbers.

$2 x^{2} + 2 x + 5 = 0$

Solve the equations over the complex numbers.

$5 x^{2} + 6 x + 2 = 0$

Solve the equations over the complex numbers.

$x^{2} + x + 2 = 0$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(2 , 0\right) , \left(x , y\right) = \left(4 , 4\right)$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(0 , 1\right) , \left(x , y\right) = \left(2 , 5\right)$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(- 5 , 3\right) , \left(x , y\right) = \left(2 , 9\right)$

Use the vertex $\left(h , k\right)$ and a point on the graph $\left(x , y\right)$ to find the general form of the equation of the quadratic function.

$\left(h , k\right) = \left(0 , 1\right) , \left(x , y\right) = \left(1 , 0\right)$

Use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

Use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

Use a calculator to find the answer.

Graph on the same set of axes $f \left(x\right) = x^{2} , f \left(x\right) = x^{2} + 2$ and $f \left(x\right) = x^{2} , f \left(x\right) = x^{2} + 5$ and $f \left(x\right) = x^{2} - 3.$ What appears to be the effect of adding a constant?

Use a calculator to find the answer.

The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function $h \left(x\right) = \frac{- 32}{\left(80\right)^{2}} x^{2} + x$ where $x$ is the horizontal distance traveled and $h \left(x\right)$ is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

Use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $\left(1 , −2\right) ,$ opens up.

Use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex $\left(−5 , 11\right) ,$ opens down.

Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\left(1 , 1\right)$ and has shape of $f \left(x\right) = 2 x^{2} .$ Vertex is on the $y -$ axis.

Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\left(2 , 3\right)$ and has the shape of $f \left(x\right) = 3 x^{2} .$ Vertex is on the $y -$ axis.

Write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains $\left(4 , 3\right)$ and has the shape of $f \left(x\right) = 5 x^{2} .$ Vertex is on the $y -$ axis.

Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by $h \left(t\right) = - 4.9 t^{2} + 229 t + 234.$ Find the maximum height the rocket attains.

A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?