Algebra: Quadratic Functions
Contents (3 topics • 30 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 \( \big[ f(-\frac{b}{2a}), +\infty \big) \).
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 \( \big[ k, +\infty \big) \).
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
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)
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