Problem 19 - Algebra 2

Question

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=x^{2}-9 x+9 $$

Step-by-Step Solution

Answer

y-intercept: (0, 9); Axis of symmetry: x = 4.5; Vertex at (4.5, -2.25).

Step 1: Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( f(x) \).\[ f(0) = 0^2 - 9(0) + 9 = 9 \]Thus, the y-intercept is \((0, 9)\).

Step 2: Determine the axis of symmetry and x-coordinate of the vertex

For a quadratic function of the form \( f(x) = ax^2 + bx + c \), the axis of symmetry is given by \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -9 \), so: \[ x = -\frac{-9}{2 \cdot 1} = \frac{9}{2} = 4.5 \]The equation of the axis of symmetry is \( x = 4.5 \), and the \( x \)-coordinate of the vertex is \( 4.5 \).

Step 3: Calculate the y-coordinate of the vertex

Substitute \( x = 4.5 \) into the function \( f(x) \) to find the vertex's y-coordinate.\[ f(4.5) = (4.5)^2 - 9(4.5) + 9 \]Calculate \( f(4.5) = 20.25 - 40.5 + 9 = -11.25 + 9 = -2.25 \)Thus, the vertex is at \((4.5, -2.25)\).

Step 4: Create a table of values

Create a table of values for \( x \) around the vertex, ensuring to include the vertex value. Choose \( x = 2, 3, 4, 4.5, 5, 6, 7 \):\[\begin{array}{c|c} x & f(x) \\hline2 & f(2) = (2)^2 - 9(2) + 9 = 4 - 18 + 9 = -5 \3 & f(3) = (3)^2 - 9(3) + 9 = 9 - 27 + 9 = -9 \4 & f(4) = (4)^2 - 9(4) + 9 = 16 - 36 + 9 = -11 \4.5 & f(4.5) = -2.25 \5 & f(5) = (5)^2 - 9(5) + 9 = 25 - 45 + 9 = -11 \6 & f(6) = (6)^2 - 9(6) + 9 = 36 - 54 + 9 = -9 \7 & f(7) = (7)^2 - 9(7) + 9 = 49 - 63 + 9 = -5\end{array}\]

Step 5: Graph the function

Use the table of values and the identified features of the quadratic to sketch the graph. Plot the y-intercept \((0, 9)\), the vertex \((4.5, -2.25)\), and the points from the table. Draw the parabola, symmetric about the axis of symmetry \( x = 4.5 \).

Key Concepts

Vertex Form

Quadratic functions can be expressed in various forms, and one such form is the vertex form. It is especially useful when you need to quickly identify the vertex of the parabola. The vertex form is given by: \[ f(x) = a(x-h)^2 + k \]Here,

• \( a \) determines the width and orientation of the parabola—whether it opens upwards or downwards.
• \( (h, k) \) represents the vertex of the parabola, where \( h \) is the x-coordinate and \( k \) is the y-coordinate.

Understanding and converting to the vertex form helps in graphing a quadratic function more conveniently. Not only does it provide a clear picture of the vertex, but it also highlights the parabola's symmetry.

Axis of Symmetry

The axis of symmetry is a vital concept when dealing with quadratic functions, as it divides the parabola into two mirror-image halves. For any quadratic function in the form \( f(x) = ax^2 + bx + c \), the axis of symmetry can be found using the formula:\[ x = -\frac{b}{2a} \]

• The formula means that the line of symmetry cuts through the vertex of the parabola, vertically.

For example, in the function \( f(x) = x^2 - 9x + 9 \), setting the values into the formula gives us\( x = 4.5 \). This tells us that the parabola is symmetrical around the line \( x = 4.5 \). Using the axis of symmetry is helpful when graphing because it allows you to accurately reflect points across the axis, ensuring that the shape of the parabola is balanced.

Parabola

A parabola is the U-shaped graph you typically get from quadratic functions. This curve can open either upwards or downwards. Whether it opens up or down is determined by the sign of \( a \) in the expression \( f(x) = ax^2 + bx + c \):

• If \( a \) is positive, the parabola opens upwards.
• If \( a \) is negative, the parabola opens downwards.

In the exercise example, \( a = 1 \) (which is positive), meaning the parabola opens upwards. The vertex is the lowest point in such a parabola, acting like a valley. If the parabola opened downwards, the vertex would be the highest point—a peak. When graphing, especially with real-world problems like projectiles, knowing these characteristics of the parabola can help predict and analyze various behaviors, like the maximum height or the points of impact.

Y-Intercept

The y-intercept of a quadratic function is where the parabola crosses the y-axis. It is found by setting \( x = 0 \) in the function \( f(x) \) and solving for \( f(x) \). For example, in our function \( f(x) = x^2 - 9x + 9 \), substituting \( x = 0 \) gives us:\[ f(0) = 0^2 - 9(0) + 9 = 9 \]So, the y-intercept is at the point \((0, 9)\). This crucial point helps in quickly sketching a graph of the quadratic function.

• The y-intercept is often the starting point in the graph.
• It provides insight into the behavior of the parabola without any transformations.

When combined with the vertex and other critical points, like the axis of symmetry, the y-intercept serves as a valuable marker to ensure the graph's accuracy and alignment.