Problem 19 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. \(h(x)=(x-3)^{2}+2\)

Step-by-Step Solution

Answer

The vertex is (3, 2) and the axis of symmetry is x = 3.

Step 1: Identify the Form of the Quadratic Equation

The given quadratic function is in vertex form: \( h(x) = (x-3)^2 + 2 \). This form is \( h(x) = a(x-h)^2+k \), where \((h, k)\) is the vertex.

Step 2: Locate the Vertex

From the quadratic function \( h(x) = (x-3)^2 + 2 \), we identify \( h = 3 \) and \( k = 2 \). Thus, the vertex of the parabola is at the point \( (3, 2) \).

Step 3: Sketch and Label the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is \( (3, 2) \), the axis of symmetry is the line \( x = 3 \).

Step 4: Plot and Sketch the Parabola

To graph the quadratic function, plot the vertex \( (3, 2) \) on the coordinate plane. Determine additional points by substituting other values of \( x \) into the function and plotting the results. For example, choose \( x = 2 \) and \( x = 4 \) to find symmetric points around the axis of symmetry. Finally, connect these points to form a U-shaped curve.

Key Concepts

Vertex Form Equation

The vertex form of a quadratic equation is a special way of writing quadratic equations that makes it easy to identify the crucial points of a parabola. This form is written as: \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. Here, \(a\) determines the direction and the width of the parabola. If \(a\) is positive, the parabola opens upwards. Conversely, if \(a\) is negative, it opens downwards. The larger the absolute value of \(a\), the narrower the parabola. Using the vertex form allows you to easily sketch the graph by pinpointing the vertex first and then determining the parabola's orientation and spread. This form simplifies graphing and understanding the transformation of a basic quadratic equation.

Axis of Symmetry

In a parabola, the axis of symmetry is a crucial element that divides the parabola into two mirror-image halves. This line is always a vertical line for parabolas that open upwards or downwards. To find this axis, one just needs to look at the \( h \) value in the vertex form equation: - For a quadratic function \( g(x) = a(x - h)^2 + k \), the axis of symmetry is the line \( x = h \). It's vital to understand the axis of symmetry because: - It's a source of balance in the parabola, showing where it's folded in half. - Helps in plotting symmetrical points.By knowing the axis of symmetry, students can plot accurate graphs and understand how altering the equation affects the parabola's symmetry.

Parabola Vertex Identification

Identifying the vertex of a parabola is essential for graphing and understanding the quadratic function. The vertex is a turning point, representing either the maximum or minimum value of the parabola. For a parabola in vertex form: - \( y = a(x-h)^2 + k \), the vertex is simply \( (h, k) \). This identification is straightforward and provides insight into the parabola's placement on a graph: - The \( h \) value gives the \( x \)-coordinate of the vertex. - The \( k \) value gives the \( y \)-coordinate of the vertex.The vertex is crucial for graphing because it helps in accurately sketching the shape and placement of the parabola. Understanding the vertex ensures the graph starts from the correct point, making it a central part of successful graphing.