Problem 20 Complete parts a-c for each quad... [FREE SOLUTION]

Chapter 5: Problem 20

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. $$ f(x)=x^{2}-4 x-5 $$

Short Answer

Expert verified

Vertex is (2, -9), y-intercept is (0, -5), axis of symmetry is x = 2.

Step by step solution

Find the y-intercept

The y-intercept of a quadratic function is found by evaluating the function at $ x = 0 $. For the function $ f(x) = x^2 - 4x - 5 $, substitute $ x = 0 $:\[f(0) = 0^2 - 4(0) - 5 = -5\]Thus, the y-intercept is $ (0, -5) $.

Find the Axis of Symmetry Equation

The equation for the axis of symmetry of a quadratic function $ f(x) = ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $. For $ f(x) = x^2 - 4x - 5 $, we have $ a = 1 $ and $ b = -4 $:\[x = -\frac{-4}{2(1)} = 2\]So, the axis of symmetry is $ x = 2 $.

Find the x-coordinate of the Vertex

The x-coordinate of the vertex is the same as the equation of the axis of symmetry, $ x = 2 $.

Find the y-coordinate of the Vertex

To find the y-coordinate of the vertex, substitute $ x = 2 $ into the function $ f(x) = x^2 - 4x - 5 $:\[f(2) = 2^2 - 4(2) - 5 = 4 - 8 - 5 = -9\]Thus, the vertex is $ (2, -9) $.

Create a Table of Values

Create a table of values for $ x $ including the vertex $ (2, -9) $ and other points around it. Let's choose $ x = 0, 1, 2, 3, 4 $:\[\begin{array}{c|c}x & f(x) \\hline0 & -5 \1 & 1^2 - 4(1) - 5 = -8 \2 & -9 \3 & 3^2 - 4(3) - 5 = -8 \4 & 16 - 16 - 5 = -5\end{array}\]So, we have the points: $(0, -5), (1, -8), (2, -9), (3, -8), (4, -5)$.

Plot the Graph

Using the information gathered, you can plot the graph of $ f(x) = x^2 - 4x - 5 $. Plot the vertex $ (2, -9) $, the y-intercept $ (0, -5) $, and the additional points $(1, -8)$, $(3, -8)$, and $(4, -5)$. Draw a smooth curve through these points to complete the parabolic graph.

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Key Concepts

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

Vertex

The vertex of a quadratic function is a crucial component because it represents either the highest or lowest point on the graph of the function. For a quadratic function in the form \[f(x) = ax^2 + bx + c\]to find the vertex:

The x-coordinate of the vertex can be found using the formula $ x = -\frac{b}{2a} $. This formula locates the line of symmetry of the parabola.
In the given function $ f(x) = x^2 - 4x - 5 $, applying the formula, we find the x-coordinate of the vertex is $ x = 2 $.
Substitute $ x = 2 $ back into the function to find the y-coordinate; thus, $ f(2) = 2^2 - 4(2) - 5 = -9 $.

So, the vertex is $(2, -9)$. This position helps in understanding the graph's turning point, making it a point of maximum or minimum depending on the parabola's direction.

Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex, offering a simplistic view of the parabola's balance.

The formula to find the axis of symmetry for any quadratic function $ f(x) = ax^2 + bx + c $ is $ x = -\frac{b}{2a} $.
In our function, $ f(x) = x^2 - 4x - 5 $, plug in the values: $ x = -\frac{-4}{2 \times 1} = 2 $.

This information indicates that the axis of symmetry is the line $ x = 2 $. Knowing this allows us to predict that points equidistant from this line will yield equal values of $ f(x) $. It aids in creating an even plot across the graph.

Y-intercept

The y-intercept of a quadratic function is the point where the graph intersects the y-axis. This is found when the input or $ x $-value is 0.

Substitute $ x = 0 $ into the equation $ f(x) = x^2 - 4x - 5 $. Calculating gives $ f(0) = 0^2 - 4(0) - 5 = -5 $.
Thus, the y-intercept is $(0, -5)$.

This point is significant as it serves as a starting point for plotting the graph along with the vertex. The y-intercept helps in understanding how the graph behaves near the y-axis and provides a benchmark for sketching it precisely. Notably, every quadratic graph will have exactly one y-intercept.

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91影视

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. $$ f(x)=x^{2}-4 x-5 $$

Short Answer

Step by step solution

Find the y-intercept

Find the Axis of Symmetry Equation

Find the x-coordinate of the Vertex

Find the y-coordinate of the Vertex

Create a Table of Values

Plot the Graph

Key Concepts

Vertex

Axis of Symmetry

Y-intercept

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