Problem 34 - Algebra 2

Question

The graph of each function contains the given point. Find the value of \(c .\) $$ y=-5 x^{2}+c ;(2,-14) $$

Step-by-Step Solution

Answer

c = -4

Step 1: Plug in the coordinates of the given point

Substitute the x-value (2) and the y-value (-14) from the point (2, -14) into the equation to find the value of c.

Step 2: Solve for c

Replace y with -14 and x with 2 in the equation and solve for the variable c.

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in algebra and are recognizable by their typical parabolic shape when graphed. The general form of a quadratic function is given by the equation \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \), are constants and \( a \) is not equal to zero.

Understanding the role of these constants is crucial. The leading coefficient, \( a \), determines the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards. The constant \( c \) represents the y-intercept, the point where the graph crosses the y-axis.

Let's take the exercise in question: \( y = -5x^2 + c \). Since the coefficient of \( x^2 \) is negative, our parabola opens downwards, which aligns with the provided point (2, -14) lying below the x-axis. To find the value of \( c \) we must integrate the given coordinates into this general form.

Solving for Variables

Solving for variables is a common task in algebra involving finding unknown quantities, represented by letters, within mathematical equations. This detective-like work often requires manipulation of the equation to isolate the variable of interest.

For example, given the quadratic equation from our exercise, \( y = -5x^2 + c \), we are tasked with finding the value of \( c \). To do so, we substitute known values into the equation and then solve for \( c \) using algebraic methods, such as addition, subtraction, multiplication, and division.

In the case of our exercise, we follow these steps:

• Step 1: Substitute the known values from the point (2, -14) into the equation, replacing \( x \) with 2 and \( y \) with -14.
• Step 2: Solve for \( c \) by simplifying the equation. \( -14 = -5(2)^2 + c \)

Our goal is to isolate \( c \), thus translating the abstract into the concrete.

Function Graphs

Function graphs are visual representations of equations on a coordinate plane. They provide valuable insights into the behavior of functions such as their increasing or decreasing trends, intercepts, and symmetries.

For quadratic functions, the graph is a parabola. Analyzing these graphs can help in understanding the relationship between the algebraic expression of the function and its geometric representation. Points on the graph, like the one given in our exercise (2, -14), correspond to inputs and outputs of the function.

The process of incorporating a known point to find a function's constant, involves graphing the quadratic equation and pinpointing where the coordinates intersect the parabola. Thus, visualizing the exercise, \( y = -5x^2 + c \) with the point (2, -14), can confirm the accuracy of our algebraic solution for \( c \) and deepen our understanding of the quadratic function as a whole.