Problem 21 - Precalculus : Building Concepts and Connections

Question

Factor to find the \(x\)-intercepts of the parabola described by the quadratic function. Also find the real zeros of the function. $$h(s)=-s^{2}+2 s-1$$

Step-by-Step Solution

Answer

The x-intercepts or real zeros of the function \(h(s)=-s^{2}+2 s-1\) is \(s = 1\)

Step 1: Set the Function to Zero

We start by setting the function \(h(s)\) to equal zero. This gives us the equation: \(-s^{2} + 2s - 1 = 0\)

Step 2: Rearranging the terms

To simplify factoring, we will rearrange the equation to read \(s^{2} - 2s + 1 = 0\) which is much easier to factor.

Step 3: Factor the Quadratic

We now factor the quadratic. The factored form of \(s^{2} - 2s + 1 = 0\) is \((s-1)^{2} = 0\)

Step 4: Solve for s

Setting each factor equal to zero and solving for \(s\) we get: \(s-1 = 0\) so \(s = 1\)

Key Concepts

Factoring Quadratics

When dealing with quadratic functions, one of the common tasks is factoring. This process involves rewriting the quadratic expression in a product form comprising linear factors, making it easier to solve for the unknown variables. Factoring is especially useful when you need to find roots or solutions of a quadratic equation. Given the form of a quadratic function: \[ ax^2 + bx + c = 0 \] The goal is to transform it into: \[(x - n_1)(x - n_2) = 0 \] Where \( n_1 \) and \( n_2 \) are the solutions or roots.To factor the quadratic \( s^2 - 2s + 1 \), we look for two numbers that add to -2 (the coefficient of \( s \)) and multiply to +1 (the constant term). These numbers are both -1. Hence, the quadratic can be expressed as: \[(s - 1)(s - 1) = 0 \] Which simplifies to \((s-1)^2 = 0\). This kind of quadratic is known as a perfect square, where both roots are identical.

Real Zeros

The real zeros of a quadratic function are the values of the variable that make the function equal to zero. In the context of the function described by the equation \(-s^2 + 2s - 1\), solving for real zeros involves finding solutions for \(s\) after setting the function to zero.To find the real zeros:

• We set the function to zero: \( h(s) = 0 \) becomes \(-s^2 + 2s - 1 = 0\).
• We multiply the entire equation by -1 for easier factoring, getting \(s^2 - 2s + 1 = 0\).
• After factoring, as demonstrated before, we obtain \((s-1)^2 = 0\).
• This results in \(s = 1\) when solving \(s-1 = 0\).

Thus, the real zero of \( h(s) \) is \( s = 1 \), which indicates that the graph of the function touches or intersects the \(x\)-axis at this point.

X-Intercepts

The \(x\)-intercepts of a quadratic function are the points where the graph of the function crosses or touches the \(x\)-axis. At these points, the value of the function is zero. These intercepts are essentially the real zeros of the function.Let's see this concept in the context of the function \(h(s) = -s^2 + 2s - 1\):

• We've already converted the equation to \(s^2 - 2s + 1 = 0\) through rearranging and multiplying by -1 for easier factoring.
• Factoring results directly in \((s - 1)^2 = 0\), giving us the sole solution \(s = 1\).

The value \(s = 1\) indicates an \(x\)-intercept at point \( (1, 0) \) on the graph. This is a special type of intercept called a double root, or a repeated root, where the graph only touches the \(x\)-axis rather than fully crossing it.