Problem 17 The graph of the function \(y=x^... [FREE SOLUTION] (2024)

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Chapter 1: Problem 17

The graph of the function $y=x^{2}$ is translated to an image parabola withzeros 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

Expert verified

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

These are the key concepts you need to understand to accurately answer the question.

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.

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$Problem 17 The graph of the function \(y=x^... [FREE SOLUTION] (3)$

Most popular questions from this chapter

Alison often sketches her quilt designs on a coordinate grid. The coordinatesfor a section of one her designs are $A(-4,6)$ $\mathrm{B}(-2,-2),\mathrm{C}(0,0), \mathrm{D}(1,-1),$ and $\mathrm{E}(3,6)$ She wants to transform the original design by a horizontal stretch about the$y$ -axis by a factor of $2,$ a reflection in the $x$ -axis, and a translationof 4 units up and 3 units to the left. a) Determine the coordinates of the image points, $\mathrm{A}^{\prime},\mathrm{B}^{\prime}, \mathrm{C}^{\prime}, \mathrm{D}^{\prime},$ and$\mathrm{E}^{\prime}$. b) If the original design was defined by the function $y=f(x),$ determine theequation of the design resulting from the transformations.If the point (10,8) is on the graph of the function $y=f(x),$ what point mustbe on the graph of each of the following? a) $y=f^{-1}(x+2)$ b) $y=2 f^{-1}(x)+3$ c) $y=-f^{-1}(-x)+1$Describe what happens to the graph of a function $y=f(x)$ after the followingchanges are made to its equation. a) Replace $x$ with $4 x .$ b) Replace $x$ with $\frac{1}{4} x$ c) Replace $y$ with $2 y$ d) Replace $y$ with $\frac{1}{4} y$ e) Replace $x$ with $-3 x$ f) Replace $y$ with $-\frac{1}{3} y$Use translations to describe how the graph of $y=\frac{1}{x}$ compares to thegraph of each function. a) $y-4=\frac{1}{x}$ b) $y=\frac{1}{x+2}$ c) $y-3=\frac{1}{x-5}$ d) $y=\frac{1}{x+3}-4$Match the function with its inverse. Function a) $y=2 x+5$ b) $y=\frac{1}{2} x-4$ c) $y=6-3 x$ d) $y=x^{2}-12, x \geq 0$ e) $y=\frac{1}{2}(x+1)^{2}, x \leq-1$ Inverse $\mathbf{A} \quad y=\sqrt{x+12}$ $\mathbf{B} \quad y=\frac{6-x}{3}$ $\mathbf{c} \quad y=2 x+8$ $\mathbf{D} \quad y=-\sqrt{2 x}-1$ $\mathbf{E} \quad y=\frac{x-5}{2}$

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