Problem 27 - Algebra 1

Question

Sketch the graph of the inequality. $$y

Step-by-Step Solution

Answer

The solution to the inequality \(y

Step 1 - Determine the vertex of the parabola

The parabola described by \(y=x^{2}+x+8\) can be rewritten as \(y=(x+0.5)^{2}+7.75\). The vertex of this parabola is \(-0.5, 7.75\).

Step 2 - Draw the graph of the parabola

Sketch the parabola with the vertex at \(-0.5, 7.75\). Note that the parabola opens upwards, as the coefficient of the \(x^{2}\) term is positive.

Step 3 - Identify the solution to the inequality

Since the inequality is \(y

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. The general form of a parabolic equation is given by \( y = ax^2 + bx + c \). The sign of the coefficient \( a \) determines the direction in which the parabola opens:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In our example, the equation \( y = x^2 + x + 8 \) has \( a = 1 \), which means the parabola opens upwards. Parabolas are symmetrical about their vertex, and have only one turning point. Understanding the properties of parabolas is key to solving inequalities that involve them.

Vertex Form

The vertex form of a quadratic equation makes it easy to identify the vertex of the parabola. It is typically written as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. This form is helpful because it clearly indicates the location of the vertex, and how the graph is shifted horizontally and vertically from the origin.
In our example, the standard form \( y = x^2 + x + 8 \) is rewritten in vertex form as \( y = (x + 0.5)^2 + 7.75 \). This reveals the vertex to be at \( (-0.5, 7.75) \), which is crucial for graphing. Calculating the vertex might involve completing the square, which is a process of rewriting a quadratic into its vertex form.

Graph Sketching

Graph sketching involves drawing the graph based on its characteristics, such as its vertex and the direction in which it opens. When graphing a quadratic inequality like \( y < x^2 + x + 8 \), you first graph the equality \( y = x^2 + x + 8 \).

• Start by plotting the vertex, \( (-0.5, 7.75) \).
• Next, draw the parabolic curve through the vertex, opening upwards, since the coefficient of \( x^2 \) is positive.
• Finally, to represent the inequality \( y < x^2 + x + 8 \), shade the region below the curve. This represents all the points \( (x, y) \) that satisfy the inequality.

Graph sketching provides a visual means to solve and understand inequalities and equations. It's important to accurately represent the vertex and curve, as this forms the basis for identifying the solution region.

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Problem 27

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Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. The general form of a parabolic equation is given by \( y = ax^2 + bx + c \). The sign of the coefficient \( a \) determines the direction in which the parabola opens:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In our example, the equation \( y = x^2 + x + 8 \) has \( a = 1 \), which means the parabola opens upwards. Parabolas are symmetrical about their vertex, and have only one turning point. Understanding the properties of parabolas is key to solving inequalities that involve them.

Vertex Form

The vertex form of a quadratic equation makes it easy to identify the vertex of the parabola. It is typically written as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. This form is helpful because it clearly indicates the location of the vertex, and how the graph is shifted horizontally and vertically from the origin.
In our example, the standard form \( y = x^2 + x + 8 \) is rewritten in vertex form as \( y = (x + 0.5)^2 + 7.75 \). This reveals the vertex to be at \( (-0.5, 7.75) \), which is crucial for graphing. Calculating the vertex might involve completing the square, which is a process of rewriting a quadratic into its vertex form.

Graph Sketching

Graph sketching involves drawing the graph based on its characteristics, such as its vertex and the direction in which it opens. When graphing a quadratic inequality like \( y < x^2 + x + 8 \), you first graph the equality \( y = x^2 + x + 8 \).

• Start by plotting the vertex, \( (-0.5, 7.75) \).
• Next, draw the parabolic curve through the vertex, opening upwards, since the coefficient of \( x^2 \) is positive.
• Finally, to represent the inequality \( y < x^2 + x + 8 \), shade the region below the curve. This represents all the points \( (x, y) \) that satisfy the inequality.

Graph sketching provides a visual means to solve and understand inequalities and equations. It's important to accurately represent the vertex and curve, as this forms the basis for identifying the solution region.