Question

Determine which of the given points are on the graph of the equation \(y=3 x^{2}-8 \sqrt{x}\). Points: (-1,-5),(4,32),(9,171)

Short answer

(4, 32) is on the graph.

Step-by-step solution

- Evaluate the equation for point (-1, -5)

Substitute the point (-1, -5) into the equation \(y = 3x^{2} - 8\sqrt{x}\). We get: \(y = 3(-1)^{2} - 8\sqrt{-1}\). Calculating this: \(y = 3(1) - 8\sqrt{-1}\). The term \(\sqrt{-1}\) is not a real number. Hence, point (-1, -5) is not on the graph.

- Evaluate the equation for point (4, 32)

Substitute the point (4, 32) into the equation \(y = 3x^{2} - 8\sqrt{x}\). We get: \(y = 3(4)^{2} - 8\sqrt{4}\). Calculating this: \(y = 3(16) - 8(2) = 48 - 16 = 32\). Since the calculated value matches the given y-value, point (4, 32) is on the graph.

- Evaluate the equation for point (9, 171)

Substitute the point (9, 171) into the equation \(y = 3x^{2} - 8\sqrt{x}\). We get: \(y = 3(9)^{2} - 8\sqrt{9}\). Calculating this: \(y = 3(81) - 8(3) = 243 - 24 = 219\). Since the calculated value (219) does not match the given y-value (171), point (9, 171) is not on the graph.

Key concepts

Quadratic Equations

Quadratic equations are polynomials of degree 2. The general form is given by \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the provided exercise, the quadratic component is \(3x^2\). This indicates that the quadratic term significantly influences the curvature of the graph. Quadratic graphs are known as parabolas. They can open upwards (\(a > 0\)) or downwards (\(a < 0\)). In this case, the coefficient 3 is positive, so the parabola opens upwards.

Understanding the behavior of quadratics helps in predicting the graph's shape and how it reacts to different values of \(x\). The presence of additional terms, like \(-8\sqrt{x}\) in this function, slightly complicates matters but primarily shifts the graph.

Integer and Real Numbers

Both integer and real numbers are essential in evaluating mathematical functions. Integers are whole numbers without fractions, and they can be positive, negative, or zero. Real numbers include all rational and irrational numbers, essentially covering any number that can appear on the number line.

When substituting points into equations, as in the given task, paying attention to the nature of the numbers is important. For instance, the term \(\sqrt{-1}\) does not yield a real number, making the point (-1, -5) invalid for the equation as real numbers alone are considered in most high school-level contexts. This distinction is crucial, especially when involving square roots, as negative numbers under the square root are not real.

Function Evaluation

Function evaluation involves substituting specific values into a function to check the output. In our given exercise, we substitute the point values into the quadratic equation \(y = 3x^2 - 8\sqrt{x}\). If the calculated \(y\)-value matches the given \(y\)-coordinate of the point, the point lies on the graph.

For instance, substituting (4, 32):

• First, substitute \(x = 4\): \(y = 3(4)^2 - 8\sqrt{4}\)
• Simplify it step-by-step: \(y = 3(16) - 8(2) = 48 - 16 = 32\).

The calculated \(y\)-value equals 32, matching the given \(y\)-coordinate, so the point (4, 32) is on the graph. Function evaluation is a validation process ensuring specific points satisfy the equation.

Square Roots

Square roots of a number \(x\) are values that, when multiplied by themselves, result in \(x\). For instance, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\). It's essential to remember that square roots have two values: positive and negative, but only the positive values are considered in basic algebra unless specified otherwise.

In the given problem, the term \(-8\sqrt{x}\) indicates that the evaluation must consider the positive root. However, squares and square roots follow specific rules that must be heeded. For example, \(\sqrt{-1}\) is not a real number but rather an imaginary one. This concept was evident when evaluating point (-1, -5) in the exercise, where \(\sqrt{-1}\) couldn't be used in the context of real numbers.