Problem 20 - 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. $$f(t)=-t^{2}+4 t$$

Step-by-Step Solution

Answer

The \(x\)-intercepts of the parabola are \(t=0\) and \(t=4\). These are also the real zeros of the function.

Step 1: Write the function

Our function is \(f(t)=-t^{2}+4 t\)

Step 2: Set function equal to zero

To find the \(x\)-intercepts, setting the function equal to zero gives us \(0=-t^{2}+4t\)

Step 3: Rearrange the function

To make the function easier to factor, we can rearrange the function in the form \(at^2 + bt + c = 0\), resulting in \(t^{2}-4t=0\)

Step 4: Factor the function

The function can be factored as \(t(t-4) = 0\)

Step 5: Solve for \(t\)

Setting each factor equal to zero gives us \(t=0\) and \(t=4\), the \(x\)-intercepts of the function.

Step 6: Check for real zeros

The real zeros of the function are also \(t=0\) and \(t=4\) because these are the \(x\)-values at which the function equals zero.

Key Concepts

Understanding X-Intercepts in Quadratic Functions

In the world of quadratic functions, the x-intercepts are the points where the graph of the function crosses the x-axis. Imagine a curve or parabola sitting on a graph, and the x-intercepts are like the points where the curve touches the horizontal line of the x-axis. These points are super important because they indicate the solutions to the equation when the function equals zero. For the function given by

• \(f(t) = -t^2 + 4t\), it is set equal to zero, resulting in \(0 = -t^2 + 4t\).

To find the intercepts, you solve the equation. These solutions, \(t = 0\) and \(t = 4\), are the x-intercepts of the quadratic function. In other words, if you were to draw the graph of this function, it would cross the x-axis at the points \((0, 0)\) and \((4, 0)\).
Finding the x-intercepts is helpful as they give us the roots of the equation, offering insights into the graph's behavior.

Simplifying Quadratic Equations through Factorization

Factorization is a nifty method used to simplify quadratic equations and find solutions. It involves breaking down a complicated expression into simpler parts (or factors) that multiply together to give the original one. For our quadratic function

• \(-t^2 + 4t\), we can rearrange it as \(t^2 - 4t\) and then find the factors.

As one of the core techniques in algebra, factorization simplifies the process of solving quadratic equations by converting them into a product of easier equations. In the given exercise, the expression

• \(t(t - 4) = 0\) is the factored form of the rearranged equation, \(t^2 - 4t\).

The beauty of factorization is that it makes solving for t a breeze. By setting each factor equal to zero, \(t=0\) and \(t=4\), you can directly find the roots or solutions. This ease of direction highlights why factorization is a clever method used extensively in mathematics.

Understanding Real Zeros of Quadratic Functions

Real zeros are the values that satisfy the quadratic equation, making the function equal to zero. These zeros are crucial as they are the solutions to the equation and correspond to the x-intercepts of the function's graph. In our exercise, the real zeros are found by solving the factored form,

• \(t(t - 4) = 0\), resulting in \(t = 0\) and \(t = 4\).

These results tell us that there are two values of t for which the function, \(f(t) = -t^2 + 4t\), equals zero. These are also the points where the parabola touches the x-axis, confirming that they are indeed the x-intercepts. Real zeros have both theoretical and practical significance, as they inform us about the function's transition points on a graph and help us understand the overall shape and direction of the parabola.