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

Only the point (4, 32) lies on the graph.

Step-by-step solution

- Verify the first point

Substitute (x = -1) and (y = -5) into the equation (y = 3x^2 - 8dsqrt{x}) and check if the equation holds true.Calculate as follows: (y = 3(-1)^2 - 8dsqrt{-1}). Since (dx=-1) under the square root is undefined, the point (-1,-5) is not on the graph.

- Verify the second point

Substitute (x = 4) and (y = 32) into the equation (y = 3x^2 - 8dsqrt{x}) and check if the equation holds true.Calculate as follows: (y = 3(4)^2 - 8dsqrt{4}) = 3(16) - 8(2) = 48 - 16 = 32. The equation holds true, so the point (4, 32) is on the graph.

- Verify the third point

Substitute (x = 9) and (y = 171) into the equation (y = 3x^2 - 8dsqrt{x}) and check if the equation holds true.Calculate as follows: (y = 3(9)^2 - 8dsqrt{9}) = 3(81) - 8(3) = 243 - 24 = 219. Since (y= 219not amount (174of line (171), This make the point does not on the graph.

Key concepts

Graphing Equations

Graphing equations is a method to visualize mathematical relationships. It involves plotting points on a coordinate plane to represent the solutions of an equation. The coordinate plane has two axes: the x-axis (horizontal) and y-axis (vertical). Each point on the plane has an x-coordinate and a y-coordinate.
To determine if a point lies on the graph of an equation, substitute the x and y values of the point into the equation.

• If both sides of the equation are equal after substitution, the point is on the graph.
• If they are not equal, the point is not on the graph.

In our example, the equation is: \(y = 3x^2 - 8\sqrt{x}\). We tested three points, substituting their x and y coordinates to check if they satisfied the equation.

For instance, when we substituted \((-1, -5)\), we encountered an issue because the square root of a negative number is undefined in real numbers. Thus, this point is not on the graph. On the other hand, substituting \((4, 32)\) into the equation resulted in a true statement, indicating this point is on the graph.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest degree of the variable (x) is 2. The general form of a quadratic function is \(y = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants.
Quadratic functions graph as parabolas, which are U-shaped curves that can open upwards or downwards:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

In our equation \(y = 3x^2 - 8\sqrt{x}\), the term \(3x^2\) indicates it's a quadratic component. The coefficient 3 (a = 3) means the parabola will open upwards and be narrower compared to a parabola where a is smaller. The quadratic term dominates the graph's shape, affecting how steeply the parabola rises or falls.

Understanding the quadratic portion helps in knowing how the graph behaves over different ranges of x. This understanding can aid in making sense of why certain points do or do not lie on the function's graph.

Square Roots

The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). It is denoted as \(\sqrt{x}\). For example, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\).
Important points about square roots:

• Square roots of positive numbers are defined in the set of real numbers.
• Square roots of negative numbers are not defined in real numbers (they involve imaginary numbers).

In our equation \(y = 3x^2 - 8\sqrt{x}\), the term \(-8\sqrt{x}\) involves a square root.
When plugging into this equation, the x-value must be non-negative (\(x \geq 0\)) because square roots of negative numbers are not considered within real-valued functions.

Inserting an x-value like -1 into \(\sqrt{x}\) causes issues and indicates why certain points, such as (-1, -5), do not fit within the graph. Recognizing where square root terms apply and understanding their domain constraints is essential in working with and graphing these types of equations.