Question

Determine which of the given points are on the graph of the equation. $$ \begin{array}{l} \text { Equation: } y=x^{4}-\sqrt{x} \\ \text { Points: }(0,0) ;(1,1) ;(2,4) \end{array} $$

Short answer

Only the point \((0,0)\) is on the graph.

Step-by-step solution

Understand the Equation

The equation given is \( y = x^4 - \sqrt{x} \). This relationship provides the value of \(y\) for any given \(x\). To determine if a point \((x, y)\) lies on the graph, substitute the \(x\)-value into the equation and see if it yields the corresponding \(y\)-value.

Test the First Point \((0,0)\)

Substitute \( x = 0 \) into the equation. This gives \( y = 0^4 - \sqrt{0} = 0 - 0 = 0 \). Since \(y\) also equals 0, the point \((0,0)\) is on the graph.

Test the Second Point \((1,1)\)

Substitute \( x = 1 \) into the equation. This gives \( y = 1^4 - \sqrt{1} = 1 - 1 = 0 \). Since \(y\) should equal 1 but we get 0, the point \((1,1)\) is not on the graph.

Test the Third Point \((2,4)\)

Substitute \( x = 2 \) into the equation. This gives \( y = 2^4 - \sqrt{2} = 16 - \sqrt{2} \). Calculating this with \( \sqrt{2} \approx 1.41 \), we get \( y \approx 16 - 1.41 = 14.59 \). Since \(y eq 4\), the point \((2,4)\) is not on the graph.

Key concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents and coefficients. In our equation, we see the term \(x^4\), which is a polynomial because it involves the variable \(x\) raised to the power of 4. Polynomial functions can have constants, variables, and exponents, and they follow rules of arithmetic like addition, subtraction, and multiplication. They are often written in the general form \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\). Key aspects of polynomial functions include:

• **Degree**: The highest power of \(x\). For \(x^4\), it’s 4.
• **Coefficients**: Numbers before the variables. Here, the coefficient of \(x^4\) is 1.
• **Constant term**: Number without a variable. In our case, it's not present.

Polynomials are found everywhere, from simple algebra to advanced calculus. They form the backbone of many mathematical models used in science and engineering.

Graphing Equations

Graphing equations helps in visually understanding the relationship between variables. The equation given is \(y = x^4 - \sqrt{x}\). To find if a point lies on the graph, substitute the \(x\)-value into the equation and see if the resulting \(y\)-value matches the given point's \(y\). For example:

• For point (0,0), substituting \(x = 0\) gives \(y = 0^4 - \sqrt{0} = 0\). Thus, (0,0) is on the graph.
• For point (1,1), substituting \(x = 1\) gives \(y = 1^4 - \sqrt{1} = 0\), which is different from 1, so (1,1) is not on the graph.
• For point (2,4), substituting \(x = 2\) gives \(y = 2^4 - \sqrt{2} \approx 14.59\), which does not match 4, so (2,4) is not on the graph.

When graphing,:

• Start by plotting points.
• Connect points smoothly.
• Look for important features like intercepts and turning points.

This process helps in understanding the behavior of the graph over different values of \(x\) and helps in solving various algebraic problems.

Square Roots

The square root function is an inverse operation to squaring. In the equation, \( \sqrt{x} \) represents the positive square root of \(x\). It’s essential to remember several things about square roots:

• **Definition**: \( \sqrt{x} \) is a number that, when squared, gives \(x\). For example, \( \sqrt{4} = 2 \) because \(2^2 = 4\).
• **Non-negativity**: For real numbers, square roots are always non-negative, meaning \( \sqrt{x} \geq 0 \).
• **Properties**: They follow rules such as \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \), and \( \sqrt{a^2} = |a|\).

In our equation, \( \sqrt{x} \) affects the \(y\)-value. For instance, when substituting \( x = 2 \) into the equation \(y = x^4 - \sqrt{x}\), we get \(y \approx 16 - 1.41 = 14.59\). Understanding square roots is crucial for solving many algebra problems, especially those involving quadratic equations and their graphs.