Problem 13 - Introductory Algebra for College Students

Question

Find the \(y\) -intercept for the parabola whose equation is given. $$y=-x^{2}+8 x-12$$

Step-by-Step Solution

Answer

The y-intercept of the given parabola is -12.

Step 1: Insert value of x at the y-intercept

At the y-intercept, the value of \(x\) is zero. Therefore, \(x = 0\) is substituted into the equation \(y = -x^{2} + 8x - 12\).

Step 2: Solve for y

After plugging in \(x = 0\), the equation becomes \(y = -0^{2} + 8*0 -12\) which simplifies to \(y = -12\). So, when \(x = 0\), \(y = -12\). Thus, the y-intercept for the given equation is -12.

Key Concepts

Parabola Equation

Understanding the equation of a parabola is crucial when analyzing its graph. A standard parabola equation is represented in the form of a quadratic function:
\[ y = ax^2 + bx + c \]
where \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. The shape of the parabola—whether it opens up or down—is determined by the sign of \( a \). If \( a \) is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards. In the given example, \( y = -x^2 + 8x - 12 \), since \( a \) (the coefficient of \( x^2 \)) is negative, it indicates that the parabola opens downwards.
This is a crucial concept as it helps us visualize the graph and predict certain properties, such as the position of the vertex and the direction in which the parabola moves as \( x \) increases or decreases.

Intercepts of a Graph

Intercepts are points where the graph of an equation crosses the x-axis or y-axis. They are fundamental in understanding the behavior of functions and in graphing. There are two types of intercepts:

• X-intercept: The point(s) where the graph crosses the x-axis, found by setting \( y=0 \) and solving for \( x \).
• Y-intercept: The point where the graph crosses the y-axis, found by setting \( x=0 \) and solving for \( y \).

In the context of our exercise, we are interested in the y-intercept of the parabola. This is calculated by setting the \( x \) value to zero and solving for \( y \). It represents the point at which the parabola crosses the y-axis and is a handy reference point when sketching the graph. Knowing that the given parabola equation \( y = -x^2 + 8x - 12 \) has a y-intercept at \( y = -12 \) helps in plotting the parabola and understanding its position in relation to the coordinate axis.

Quadratic Functions

A quadratic function is a second-degree polynomial function of the form \( y = ax^2 + bx + c \), with \( a \), \( b \), and \( c \) being real numbers, and \( a \) not equal to zero. The graph of a quadratic function is a parabola, which can either open upwards or downwards. Key features of a quadratic function include:

• Vertex: The highest or lowest point on the graph, depending on the parabola's direction.
• Axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
• Intercepts: Points where the parabola crosses the axes.

Solving quadratic functions often involves finding the roots or zeroes of the function, the y-intercept, and analyzing the vertex's coordinates. In the case of our given function \( y = -x^2 + 8x - 12 \), we used the fact that the y-intercept occurs when \( x = 0 \) to find that the y-intercept is -12. Recognizing these fundamental aspects of quadratic functions can dramatically simplify the process of graphing them and understanding their behavior across the coordinate plane.