HS Math Solutions - Algebra: Quadratic Functions

Algebra: Quadratic Functions

Contents (3 topics • 30 videos)

Standard Form of a Quadratic Function (12 videos)

Vertex Form of a Quadratic Function (8 videos)

Solving Quadratic Equations (10 videos)

Standard Form of a Quadratic Function

Introduction Videos

Characteristics of Quadratic Functions in Standard Form

Theorems and Definitions

Characteristics of a Quadratic Function in Standard Form A quadratic function is written in standard form if

$f(x)=ax^2+bx+c$ where

$a$ ,

$b$ , and

$c$ are real numbers and

$a \ne 0$ .

The graph of a quadratic function is a parabola that opens upward if

$a \gt 0$ , or downward if

$a \lt 0$ .

The vertex of the parabola is the ordered pair

$V\Bigg( -\frac{b}{2a}, \text{ } f \Big( -\frac{b}{2a} \Big) \Bigg).$ The axis of symmetry is a vertical line that mirrors the two legs of the parabola. The axis of symmetry passes through the vertex and has an equation

$x=-\frac{b}{2a} .$ The vertex is a maximum value of the function if

$a \lt 0$ and a minimum value of the function if

$a \gt 0$ .

The domain of a quadratic function is all real numbers, or

$( -\infty, +\infty )$ , in interval notation.

If

$a \lt 0$ , then the range of a quadratic function is

$\big( -\infty, f( -\frac{b}{2a} ) \big]$ .

However, if

$a \gt 0$ , then the range of a quadratic function is

$\big[ f(-\frac{b}{2a}), +\infty \big)$ .

Video Practice Problems Playlist on YouTube

Characteristics of a quadratic function in standard form.

$f(x)=\frac{1}{2}x^2-5x+9$

Convert the quadratic function from vertex form to standard form.

$f(x)=(2x-5)^2-8$

Evaluate the function at the given

$x$ -values.

$f(x)=3x^2-7x+2$

$x=1 \quad x=\frac{1}{3} \quad x=0.75 \quad x=\sqrt{6}$

Evaluate the function at the given

$x$ -values.

$g(x)=-\frac{3}{4}x^2-\frac{3}{2}x+\frac{1}{3}$

$x=-2 \quad x=\frac{8}{3} \quad x=-0.5 \quad x=\frac{\sqrt{13}}{3}-1$

Determine values for

$m$ and

$n$ if the vertex of the parabola is

$V(3, -7)$ .

$f(x)=-2x^2+6nx+m-7n$

Determine values for

$m$ and

$n$ if the vertex of the parabola is

$V(-2, 2)$ .

$g(x)=-\frac{1}{4}mx^2+2nx+n$

Determine the

$x$ - and

$y$ -intercepts of the quadratic function.

$f(x)=-3(x+2)(5x-3)$

Determine the

$x$ - and

$y$ -intercepts of the quadratic function.

$f(x)=2x^2+x-15$

Write a quadratic function whose graph passes through the set of points:

$(-2, 5) \quad (2, 3) \quad (-4, 15)$

Write a quadratic function gives its roots.

$x=5, -2$

Write a quadratic function given its roots.

$x=-\frac{3}{5}, \frac{1}{3}$

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)

Vertex Form of a Quadratic Function

Theorems and Definitions

Characteristics of a Quadratic Function in Vertex Form A quadratic function is written in vertex form if

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

$a$ ,

$h$ , and

$k$ are real numbers and

$a \ne 0$ .

The graph of a quadratic function is a parabola that opens upward if

$a \gt 0$ , or downward if

$a \lt 0$ .

The vertex of the parabola is the ordered pair

$(h, k)$ .

The axis of symmetry is a vertical line that mirrors the two legs of the parabola. The axis of symmetry passes through the vertex and has an equation

$x=h$ .

The vertex is a maximum value of the function if

$a \lt 0$ and a minimum value of the function if

$a \gt 0$ .

The domain of a quadratic function is all real numbers, or

$( -\infty, +\infty )$ , in interval notation.

If

$a \lt 0$ , then the range of a quadratic function is

$\big( -\infty, k \big]$ .

However, if

$a \gt 0$ , then the range of a quadratic function is

$\big[ k, +\infty \big)$ .

Video Practice Problems Playlist on YouTube

Characteristics of a quadratic function in vertex form.

$f(x)=-(x-4)^2-2$

Evaluate the function at the given

$x$ -values.

$f(x)=-\frac{7}{2}(x-6)^2-2$

$x=-2 \quad x=-\frac{3}{4} \quad x=-0.4 \quad x=3+\sqrt{2}$

Determine the

$x$ - and

$y$ -intercepts of the quadratic function.

$f(x)=(x-7)^2+1$

Determine the

$x$ - and

$y$ -intercepts of the quadratic function.

$f(x)=3(x+5)^2-27$

Determine the quadratic function whose graph has a vertex at

$V(3, 1)$ and passes through

$P(0, 10)$ .

Determine the quadratic function whose graph has a vertex at

$V(12, 7)$ and passes through

$P(4, -9)$ .

Solve.

$3(x+6)^2-108=0$

Determine the inverse function for

$f(x)=-(x-5)^2+2$

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)

Solving Quadratic Equations

Introduction Videos

Solving Quadratic Equations by Using Square Roots

Solving Quadratic Equations by Factoring

Video Practice Problems Playlist on YouTube

Solve by factoring.

$10x^2+19x+6=0$

Solve by factoring.

$3x^2=x+14$

Solve by factoring.

$100x-4x^3=0$

Solve by factoring.

$4x(x+3)+2(x-2)+x=0$

Solve by factoring.

$(x-2)^2(x+1)=x(x+3)^2$

Solve by factoring.

$\frac{4x}{x+7} = \frac{10}{x-2}$

For the equation

$x^2+kx-12=0$ what value(s) of

$k$ will result in integer solutions?

Solve using a

$u$ -substitution.

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

Documents

Practice Problems (.PDF)

Practice Problems - Solutions (.PDF)