Problem 17 Complete parts a-c for each quad... [FREE SOLUTION]

Chapter 5: Problem 17

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. $$ f(x)=3 x^{2}+1 $$

Short Answer

Expert verified

The y-intercept is 1, axis of symmetry is $x = 0$, and the vertex is $(0, 1)$.

Step by step solution

Find the y-intercept

To find the $y$-intercept, we set $x = 0$ in the equation of the function. This gives $f(0) = 3(0)^2 + 1 = 1$. Therefore, the $y$-intercept is $1$, so the point is $(0, 1)$.

Determine the Axis of Symmetry

The axis of symmetry for a quadratic function $f(x) = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. Here, $a = 3$ and $b = 0$, so the axis of symmetry is $x = -\frac{0}{2 \times 3} = 0$. Thus, the equation of the axis of symmetry is $x = 0$.

Find the x-coordinate of the Vertex

The $x$-coordinate of the vertex is the same as the axis of symmetry, which is $x = 0$.

Make a Table of Values

To make a table of values, choose $x$ values around the vertex ($x = 0$), calculate the corresponding $y$ values, and include the vertex point.- $x = -1$, $f(-1) = 3(-1)^2 + 1 = 4$- $x = 0$, $f(0) = 1$ (Vertex)- $x = 1$, $f(1) = 3(1)^2 + 1 = 4$| $x$ | $f(x)$ ||---|---|| -1 | 4 || 0 | 1 || 1 | 4 |

Graph the Function

Plot the points from the table of values: $(-1, 4)$, $(0, 1)$, and $(1, 4)$. Draw the parabola opening upwards from the vertex at $(0, 1)$. The axis of symmetry, $x = 0$, shows the line about which the parabola is symmetric.

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Key Concepts

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

y-intercept

The y-intercept of a quadratic function is found where the graph crosses the y-axis. At this point, the value of x is always 0. To find the y-intercept, simply substitute 0 for x in the function equation. For our function, $f(x) = 3x^2 + 1$, substituting 0 gives:\[f(0) = 3(0)^2 + 1 = 1\]Therefore, the y-intercept is 1, and the corresponding point where the graph crosses the y-axis is at $(0, 1)$.

The y-intercept reveals where the parabola touches the y-axis.
Given by setting $x = 0$, it directly reflects the constant term in the function equation.

axis of symmetry

The axis of symmetry in a quadratic function is a vertical line that divides the parabola into two mirror-image halves. This axis is critical because it passes through the vertex, offering a balance point for the parabola. For functions in the form $f(x) = ax^2 + bx + c$, the axis of symmetry's equation is given by\[x = -\frac{b}{2a}\]In the example function $f(x) = 3x^2 + 1$, the values are $a = 3$ and $b = 0$. Substituting these into the formula we get:\[x = -\frac{0}{2 \cdot 3} = 0\]

The axis of symmetry is $x = 0$, or the y-axis itself in this case.
This line helps identify the vertex's x-coordinate, aiding in the graphing process.

vertex of a parabola

The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. It represents the turning point of the graph.For a function in the standard form $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex can be found using the axis of symmetry:\[x = -\frac{b}{2a}\]With $a = 3$ and $b = 0$ for $f(x) = 3x^2 + 1$, the x-coordinate of the vertex is 0. To find the y-coordinate, substitute $x = 0$ back into the function:\[f(0) = 3(0)^2 + 1 = 1\]So, the vertex of the parabola is at the point $(0, 1)$, the lowest point in our upward-opening parabola.

The vertex can serve as a reference for graphing the parabola.
Knowing both the x and y coordinates is crucial for plotting accurately.

graphing quadratic functions

Graphing a quadratic function involves plotting points and drawing the shape of the parabola. Begin with points calculated from a table of values around the vertex.Using the function $f(x) = 3x^2 + 1$, we have calculated:- For $x = -1$, $f(-1) = 4$- For $x = 0$ (vertex), $f(0) = 1$- For $x = 1$, $f(1) = 4$Plot these points: - $(-1, 4)$- $(0, 1)$ - $(1, 4)$The plotted points should form a symmetrical shape with the axis of symmetry $x = 0$.

Connect the dots smoothly to form the parabola.
The parabola opens upwards due to a positive $a$ value in the function.
The curve passes through the vertex and reflects symmetry around the axis.

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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. $$ f(x)=3 x^{2}+1 $$

Short Answer

Step by step solution

Find the y-intercept

Determine the Axis of Symmetry

Find the x-coordinate of the Vertex

Make a Table of Values

Graph the Function

Key Concepts

y-intercept

axis of symmetry

vertex of a parabola

graphing quadratic functions

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