Problem 22 Find the intercepts of the graph... [FREE SOLUTION]

Chapter 10: Problem 22

Find the intercepts of the graph of the equation. Then sketch the graph of the equation and label the intercepts. $$y=-x^{2}+8 x-12$$

Short Answer

Expert verified

The x-intercepts are at (2,0) and (6,0), and the y-intercept is at (0,-12).

Step by step solution

Calculate the x-intercepts

To find the x-intercepts, substitute 0 for y in the equation and solve for x. This gives the equation $0=-x^{2}+8 x-12$. Solving this equation will give the x-intercepts.

Solve the quadratic equation

The quadratic equation can be solved by factoring, completing the square, or the quadratic formula. In this case, factoring is easiest: We can factor the equation like so: $0=-x^{2}+8 x-12 = -(x-2)(x-6)$. Setting each factor equal to zero gives the solutions $x=2, x=6$, which are the x-intercepts.

Find the y-intercept

To find the y-intercept, substitute 0 for x in the equation and solve for y. This gives $y=-0^{2}+8*0-12 = -12$, so the y-intercept is -12.

Sketch the graph and label the intercepts

Using the obtained intercepts and the general shape of a quadratic function, sketch a curve that passes through these points. The x-intercepts are (2,0) and (6,0), while the y-intercept is (0,-12). It's important to clearly label these points on the graph.

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Key Concepts

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

Understanding X-Intercepts in Quadratic Functions

The x-intercepts of a quadratic function are the points where the graph intersects the x-axis. At these points, the value of the function (or y-value) is zero. To find the x-intercepts, you need to set the quadratic equation equal to zero and solve for the variable x. For example, in the equation $-x^2 + 8x - 12 = 0$, you're essentially finding the values of x for which the output y becomes zero.

Identify the quadratic equation from the function, which is $-x^2 + 8x - 12 = 0$.
Solve this equation by using various methods such as factoring, completing the square, or the quadratic formula.

In our case, by factoring the equation, we recognize it can be rewritten as $-x^2 + 8x - 12 = -(x-2)(x-6)$. This indicates the x-intercepts occur when each factor equals zero: $x = 2$ and $x = 6$. Thus, the points (2,0) and (6,0) describe where the graph crosses the x-axis.

Exploring the Y-Intercept in Quadratic Functions

The y-intercept of a quadratic function is where the graph intersects the y-axis. This intersection occurs when the x-value is zero. To find the y-intercept, substitute 0 for x in the quadratic equation and solve for y. In the equation $y = -x^2 + 8x - 12$, the y-intercept is found as follows:

Set $x = 0$ to get the expression: $y = -0^2 + 8 \times 0 - 12$
This simplifies directly to $y = -12$.

Therefore, the y-intercept is the point (0,-12), indicating where the graph meets the y-axis. This point provides a clear location on the graph, assisting in drawing or understanding the shape of the quadratic curve.

The Quadratic Formula and Its Application

The quadratic formula is a mathematical solution used to find the x-intercepts of a quadratic equation when other methods are not suitable. The standard format of a quadratic equation is $ax^2 + bx + c = 0$. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is a convenient tool, especially when the quadratic doesn't factor easily. Each of the terms in the formula plays a significant role:

$a$: The coefficient of $x^2$.
$b$: The coefficient of $x$.
$c$: The constant term.

Plugging these values from the equation into the formula helps solve for x explicitly. It's important to remember that the $\pm$ sign indicates there are generally two solutions, representing the two x-intercepts of the quadratic function.

Factoring Quadratics Easily

Factoring quadratics is a method used to solve for x-intercepts by expressing the quadratic equation as a product of linear factors. Factoring is often the simplest solution when the quadratic equation is easily reducible. For example, consider the equation $0 = -x^2 + 8x - 12$, which can be reorganized into the form:

Look for two numbers that multiply to the constant term and add to the coefficient of the middle term.
The factors of $-12$ that add up to $8$ are $2$ and $6$.
This results in $-x^2 + 8x - 12 = -(x-2)(x-6)$.

Factoring splits the quadratic into two binomials, making it straightforward to find the intercepts. This method simple, leveraging basic arithmetic operations, particularly useful for quadratic equations that can be factorized without much complexity.

91影视

Find the intercepts of the graph of the equation. Then sketch the graph of the equation and label the intercepts. $$y=-x^{2}+8 x-12$$

Short Answer

Step by step solution

Calculate the x-intercepts

Solve the quadratic equation

Find the y-intercept

Sketch the graph and label the intercepts

Key Concepts

Understanding X-Intercepts in Quadratic Functions

Exploring the Y-Intercept in Quadratic Functions

The Quadratic Formula and Its Application

Factoring Quadratics Easily

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