Math 3: Midterm Practice

Front

What is the simplified form of \(3\sqrt{12}+5\sqrt{27}\)?

Back

The simplified form is \(21\sqrt{3}\).

Front

How do you solve the equation \(3-\sqrt[3]{x-7}=5\)?

Back

To solve, isolate \(\sqrt[3]{x-7}\) to get \(x=-1\).

Front

What is the general form of a cubic function?

Back

A cubic function can be expressed as \(f(x) = a(x - h)^3 + k\), where \(a\), \(h\), and \(k\) are constants.

Front

How do you find the inverse of a linear function?

Back

To find the inverse of a function \(f(x) = mx + b\), swap \(x\) and \(y\) and solve for \(y\).

Front

What is the process for simplifying square roots?

Back

To simplify square roots, factor the number under the root into perfect squares and simplify accordingly.

Front

What is the significance of the leading coefficient in a polynomial function?

Back

The leading coefficient determines the end behavior of the polynomial function.

Front

How do you determine the vertex of a quadratic function?

Back

The vertex of a quadratic function \(f(x) = ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\).

Front

What is the relationship between a function and its inverse?

Back

A function and its inverse 'undo' each other, meaning \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Front

What is the formula for the inverse of the function \(f(x) = \frac{1}{4}x - 3\)?

Back

The inverse is \(f^{-1}(x) = 4x + 12\).

Front

How do you graph a cubic function?

Back

To graph a cubic function, identify key points, including the y-intercept and any turning points, and plot them.