Problem 7 Draw the graph of a quadratic fu... [FREE SOLUTION]

Chapter 10: Problem 7

Draw the graph of a quadratic function using the given facts about its graph. \- Opens upward -Vertex: \((-1,-4)\) -(GRAPH CANT COPY) -\(y\) -intercept: \((0,-3)\) -\(x\) -intercepts: \((-3,0),(1,0)\)

Short Answer

Expert verified

The quadratic function is \( y = x^2 + 2x - 3 \).

Step by step solution

Understand the Quadratic Function Form

The general form of a quadratic function is \( ax^2 + bx + c \). The graph of a quadratic function is a parabola, and it opens upward when \( a > 0 \). Given the function opens upward, \( a \) is positive.

Use the Given Vertex

The vertex form of a quadratic function is \( a(x-h)^2 + k \), where \((h,k)\) is the vertex. Here, the vertex is \((-1, -4)\), so substituting gives: \( y = a(x + 1)^2 - 4 \).

Use the y-intercept

We can use the given \( y \)-intercept \((0, -3)\) to find \( a \). Substitute \( x = 0 \) and \( y = -3 \) into the equation: \[-3 = a(0 + 1)^2 - 4\] Solve for \( a \): \[-3 = a - 4\] \[a = 1\].

Confirm using x-intercepts

To confirm, use the given \( x \)-intercepts \((-3, 0)\) and \((1, 0)\). With \( a = 1 \), the function becomes \( y = (x + 1)^2 - 4 \). Check at \( x = -3 \): \[0 = (-3 + 1)^2 - 4 \Rightarrow 0 = 4 - 4\] Check at \( x = 1 \): \[0 = (1 + 1)^2 - 4 \Rightarrow 0 = 4 - 4\] Both checks confirm that the function is correct.

Final Equation and Graphing

The final equation of the function is \( y = (x + 1)^2 - 4 \) or \( y = x^2 + 2x - 3 \). This confirms the vertex, opens upward, and matches all the given intercepts. You can plot this equation by finding additional points or sketching based on these features.

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Key Concepts

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

Graphing Quadratic Equations

Quadratic equations can appear in two primary forms - the standard form and the vertex form. For graphing purposes, understanding either form is important. Primarily, we will focus on the standard form here. The standard form of a quadratic equation is given by: \[ ax^2 + bx + c \] Where:

a determines the direction in which the parabola opens.
b and c determine the position of the parabola on the graph.

If \(a\) is positive, the parabola opens upwards, and when \(a\) is negative, it opens downwards. To graph a quadratic equation, it helps to find certain key points such as the vertex, x-intercepts, and y-intercepts. These points provide a skeleton on which you can base the shape of the graph. Connect these points smoothly to form the characteristic 'U' shape.

Parabolas

Parabolas are the graphical representation of quadratic functions. Each quadratic function can be represented by a corresponding parabola. These curves are symmetrical, meaning they are mirror images along a vertical line called the axis of symmetry. The most important features of a parabola include its vertex, direction of opening, and intercepts.A parabola can open either up or down:

If it opens upward, the vertex is the minimum point.
If it opens downward, the vertex is the maximum point.

Parabolas that open upwards typically model real-world applications such as the path of a projectile in physics.Additionally, parabolas help in understanding how changes in the equation affect its shape. Moving the vertex or changing the value of \(a\) can help the parabola appear steeper or wider.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is very useful for graphing because it directly reveals the vertex of the parabola, making it easier to plot. It is written as: \[ y = a(x - h)^2 + k \] This expresses the quadratic function in relation to its vertex point, \((h, k)\).

h and k are the x and y coordinates of the vertex, respectively.
a still indicates the direction of the parabola's opening and its "vertical stretch" indicating steepness.

To convert from standard form to vertex form, you can complete the square or use the vertex formula. In practice, using the vertex form is often easier for graphing tasks because it requires fewer calculations to find the vertex, which serves as the focal point from which other features can be determined.

x and y Intercepts

Understanding x and y intercepts of a quadratic function is crucial for graphing and analyzing its behavior. These intercepts occur when the parabola crosses the x-axis and y-axis.**Finding the x-intercepts:**

Set the quadratic function equal to zero and solve for x.
The solutions, if real, are the points where the parabola crosses the x-axis.

In our example, the x-intercepts are given as \((-3, 0)\) and \((1, 0)\), which means these are the points where the quadratic touches or crosses the x-axis.**Finding the y-intercept:**

Substitute x = 0 into the quadratic equation.
The result is the y-value where the parabola crosses the y-axis.

In our case, the y-intercept is given as \((0, -3)\). By understanding these intercepts, you can quickly plot or analyze the basic structure and placement of the parabola on a graph.

91影视

Draw the graph of a quadratic function using the given facts about its graph. \- Opens upward -Vertex: \((-1,-4)\) -(GRAPH CANT COPY) -\(y\) -intercept: \((0,-3)\) -\(x\) -intercepts: \((-3,0),(1,0)\)

Short Answer

Step by step solution

Understand the Quadratic Function Form

Use the Given Vertex

Use the y-intercept

Confirm using x-intercepts

Final Equation and Graphing

Key Concepts

Graphing Quadratic Equations

Parabolas

Vertex Form of a Quadratic Function

x and y Intercepts

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