Problem 23 - Algebra 1: Concepts and Skills

Question

Decide whether the parabola opens up or down. $$ y=8 x-x^{2} $$

Step-by-Step Solution

Answer

The parabola opens down.

Step 1: Write the equation in standard form

The given equation is \( y=8 x - x^{2} \). It can be rewritten into standard form by flipping the terms with 'x' around: \( y = -x^2 + 8x \).

Step 2: Identify the coefficient of \( x^{2} \)

The equation is \( y = -x^{2} + 8x \), meaning that the coefficient 'a' of x² is -1.

Step 3: Determine the direction of the opening of the parabola

Because the coefficient of \( x^{2} \) is negative, this informs us that the parabola opens down.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and take the form: \[ y = ax^2 + bx + c \] In this expression, \(a\), \(b\), and \(c\) are constants, which can be any real numbers, and \(a\) should not be zero.

• The term \(ax^2\) is known as the quadratic term.
• The term \(bx\) is called the linear term.
• The term \(c\) is the constant term.

These equations graph as parabolas, which are u-shaped curves. The direction of this curve—either opening upwards or downwards—depends on several factors, particularly the value of the coefficient \(a\). Understanding these equations is key to solving problems involving parabola graphs.

Coefficients

In a quadratic equation, coefficients are the numbers that multiply the variables. For the general quadratic expression \( ax^2 + bx + c\), there are three coefficients:

• \(a\), the coefficient of the quadratic term \(x^2\)
• \(b\), the coefficient of the linear term \(x\)
• \(c\), the constant term

These coefficients shape the parabola's graph in different ways:New lines can be helpful:
* The coefficient \(a\) affects the direction and width of the parabola
* If \(a\) is positive, the parabola opens upwards, like a smile.
* If \(a\) is negative, the parabola opens downwards, like a frown. Notice the use of small changes:
The coefficients \(b\) and \(c\) also contribute by affecting the vertex and axis of symmetry of the parabola but do not influence its direction. It's crucial to identify these coefficients correctly in any quadratic equation to understand the behavior of the parabola.

Graph Direction

The direction in which a parabola opens is crucial to understanding its shape and behavior. The sign of the quadratic coefficient \( a \) in the equation \( ax^2 + bx + c \) directly determines this direction.Let's break it down:

• A positive \(a\) means the parabola opens upwards. It can be visualized as resembling a cup or a bowl. This is why it's often called a 'smiley' parabola.
• A negative \(a\) causes the parabola to open downwards. This makes it appear like an upside-down cup or a frown, thus sometimes referred to as a 'frowny' parabola.

In our given exercise problem, the quadratic equation was rearranged to \( y = -x^2 + 8x \). Here, the \(a\) coefficient is \(-1\), indicating the parabola is a 'frowny' one. Understanding the graph direction helps in solving quadratic problems as it provides insights into the parabolic path and points like the vertex and intercepts.