Digital SAT Practice Questions

The quadratic function f is defined by f(x) = x^2 - 6x + c, where c is a constant. If the graph of y = f(x) in the xy-plane has exactly one distinct real x-inte...

In the xy-plane, the graph of the equation x^2 + y^2 - 10x + 8y + k = 0 is a circle, where k is a constant. If the radius of the circle is 7, what is the value ...

\[ \frac{1}{2}x - \frac{3}{4}y = 5 \] \[ cx - 3y = 20 \] In the given system of linear equations, \( c \) is a constant. For what value of \( c \) does the syst...

A toy rocket is launched from a raised platform. Its height \( h(t) \), in feet, after \( t \) seconds is given by the function \[ h(t) = -16t^2 + 64t + 80 \] W...

If \( x > 0 \), which of the following expressions is equivalent to \[ \sqrt{x^5} \cdot \sqrt[3]{x^2} \] ?...

A culture of bacteria initially contains 500 bacteria. The population increases by 15% every 4 hours. Which of the following functions \( P(t) \) models the num...

The equation of a circle in the \( xy \)-plane is given by \[ x^2 + 8x + y^2 - 10y = 8 \] What is the radius of the circle?...

Consider the following system of linear equations: \[ 3x - 4y = 12 \] \[ 6x + ky = 10 \] In the system of equations above, \( k \) is a constant. If the system ...

The function \( f \) is defined by \( f(x) = -2x^2 + 16x - 24 \). Which of the following equivalent forms of the function \( f \) displays the maximum value of ...

The equation of a circle in the \( xy \)-plane is given by: \[ x^2 + y^2 - 10x + 6y + 9 = 0 \] If this circle is translated 2 units to the right and 3 units dow...

A conservationist is tracking a population of a specific rare beetle in a protected forest. The population is observed to increase by \( 12\% \) every year. If ...

Consider the rational equation below: \[ \frac{x^2 + 5x + 6}{x + 3} = x^2 + 2x - 4 \] What is the sum of all valid solutions \( x \) to the given equation?...

A system of linear equations is shown below, where \( a \) and \( b \) are constants. \[ \frac{1}{2}x - \frac{3}{4}y = 5 \] \[ ax - by = 20 \] If the system has...

The height \( h(t) \), in feet, of a small rocket \( t \) seconds after it is launched from a platform is modeled by the function: \[ h(t) = -16t^2 + 64t + 80 \...

A scientist is studying a radioactive isotope in a laboratory. The mass of the isotope decreases by \( 20\% \) every \( 6 \) days. If the initial mass of the is...

Which of the following represents all solutions to the rational equation below? \[ \frac{x}{x - 4} - \frac{4}{x + 4} = \frac{32}{x^2 - 16} \]...

The equation of a circle in the \( xy \)-plane is given below: \[ x^2 + y^2 - 10x + 8y - 8 = 0 \] What is the radius of the circle?...

The system of linear equations below has infinitely many solutions. \[ 5x - 7y = 15 \] \[ kx - \frac{21}{2}y = m \] If \( k \) and \( m \) are constants, what i...

The equation \( y = -3x^2 + bx + c \) represents a parabola in the \(xy\)-plane, where \( b \) and \( c \) are constants. If the vertex of the parabola is at th...

A circle in the \(xy\)-plane has its center at the origin. The total area of the circle is \( 144\pi \). A sector of the circle is bounded by radii \( OA \) and...

The population of a certain species of fish in a newly conserved lake is modeled by the function \( P(t) = 1200(0.82)^{\frac{t}{3}} \), where \( t \) is the num...

The polynomial \( p(x) = 2x^3 - 5x^2 + ax + b \), where \( a \) and \( b \) are constants, is perfectly divisible by \( (x - 2) \). When \( p(x) \) is divided b...

The system of equations is given by: \[ y = ax^2 + 4x - 5 \] \[ y = -2x + b \] where \( a \) and \( b \) are constants, and \( a > 0 \). If the system of equati...

What is the sum of all valid solutions to the given equation? \[ \frac{x^2 - 3x}{x + 2} = \frac{10}{x + 2} + 2 \]...