Question

The graph of the function \(y=x^{2}\) is translated to an image parabola with zeros 7 and \(1 .\) a) Determine the equation of the image function. b) Describe the translations on the graph of \(y=x^{2}\). c) Determine the \(y\) -intercept of the translated function.

Short answer

a) \( y = x^2 - 8x + 7 \)b) The graph is translated right by 4 units and down by 9 units.c) y-intercept is 7.

Step-by-step solution

Determine the Zeros

We need to find a quadratic function that has zeros at 7 and 1. The zeros of a quadratic function are the values of x for which the function equals zero.

Form the Quadratic Equation

Since the function has roots at 7 and 1, we can write it in factored form as: \( y = a(x-7)(x-1) \)

Expand the Quadratic Equation

Expand the expression \( y = a(x-7)(x-1) \): \[ y = a(x^2 - 8x + 7) \].

Determine the Value of 'a'

Since we are looking for a function of the form similar to \( y = x^2 \), we will assume that \( a = 1 \). Therefore, the equation simplifies to \( y = x^2 - 8x + 7 \).

Identify the Translations

The original graph of \( y = x^2 \) has been translated horizontally and vertically. The vertex form of a translated parabola is \( y = a(x - h)^2 + k \). For this problem, we see that the parabola has shifted rightward by 4 units and downward by 9 units.

Find the y-Intercept

To find the y-intercept, we set \( x = 0 \) and solve for \( y \): \( y = 0^2 - 8(0) + 7 = 7 \). So, the y-intercept is at the point (0, 7).

Key concepts

parabola translation

When we talk about translating a parabola, we mean shifting its graph horizontally or vertically on the coordinate plane. The standard form of a quadratic function is \( y = x^2 \). However, when we translate this parabola, we use the vertex form: \( y = a(x - h)^2 + k \). Here is what each element means:

• \(h\): Horizontal shift. If \(h\) is positive, the parabola moves right; if negative, it moves left.
• \(k\): Vertical shift. If \(k\) is positive, the parabola moves up; if negative, it moves down.

In our case, the zeros of the parabola were given at \(7\) and \(1\). To convert this into vertex form, we can determine the shifts applied to the original \( y = x^2 \). By factoring the expanded form, we see the shifts as detailed in step 5. Therefore, our final translated function is \( y = x^2 - 8x + 7 \). This indicates it has been translated 4 units to the right and 9 units down from the original \(y = x^2\).

quadratic function roots

Roots or zeros of a quadratic function are the x-values where the function equals zero. For our function \( y = x^2 - 8x + 7 \), the roots can be found by solving the equation \( x^2 - 8x + 7 = 0 \).

The general method to find roots involves factoring, using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), or completing the square. For the given function:

1. **Write in factored form**: Set the function to zero and solve: \( x^2 - 8x + 7 = 0 \) becomes \( (x - 7)(x - 1) = 0 \).
2. **Identify roots from factored form**: The solutions (or roots) are \( x = 7 \) and \( x = 1 \).

This tells us that the points where the parabola intersects the x-axis are at \( (7, 0) \) and \( (1, 0) \). These roots are essential for determining the shape and position of the parabola.

y-intercept

The y-intercept of a quadratic function is the point where the graph crosses the y-axis. To find this, we set \( x = 0 \) in the function and solve for \( y \). For the translated function \( y = x^2 - 8x + 7 \):
1. **Set \( x \) to 0**: \( y = 0^2 - 8(0) + 7 \)
2. **Solve for \( y \)**: \( y = 7 \).

Thus, the y-intercept is at the point \( (0, 7) \). This is an important feature of the graph because it tells us where the parabola touches the y-axis. In this case, at the point \( (0, 7) \), which verifies the vertical shift described earlier.