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Chapter 8: Problem 54

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=3 x^{2}-13 x-10 $$

Short Answer

Expert verified
The vertex is \( \left( \frac{13}{6}, -\frac{149}{12} \right) \), the graph opens upward, y-intercept is \( (0, -10) \), x-intercepts are \( (5, 0) \) and \( \left( -\frac{2}{3}, 0 \right) \).

Step by step solution

01

Identify the components of the quadratic function

The given quadratic function is \( f(x) = 3x^2 - 13x - 10 \). This is in the standard form \( ax^2 + bx + c \), where \( a = 3 \), \( b = -13 \), and \( c = -10 \).
02

Determine the direction of the parabola

Since the coefficient \( a = 3 \) is positive, the parabola opens upward.
03

Find the vertex of the parabola

The vertex \((h, k)\) of a parabola given by \( ax^2 + bx + c \) can be found using the formula \( h = -\frac{b}{2a} \). Substituting the values, we get \( h = -\frac{-13}{2 \times 3} = \frac{13}{6} \). Calculate \( k \) by finding \( f(\frac{13}{6}) \): \[f\left(\frac{13}{6}\right) = 3\left(\frac{13}{6}\right)^2 - 13\left(\frac{13}{6}\right) - 10 \]Computing this gives \( k = -\frac{149}{12} \). Therefore, the vertex is \( \left( \frac{13}{6}, -\frac{149}{12} \right) \).
04

Find the y-intercept

The y-intercept of the quadratic function is found by evaluating the function at \( x = 0 \). Substituting, we get:\[f(0) = 3(0)^2 - 13(0) - 10 = -10\]Therefore, the y-intercept is \( (0, -10) \).
05

Find the x-intercepts

To find the x-intercepts, set \( f(x) = 0 \) and solve the quadratic equation \( 3x^2 - 13x - 10 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 3 \), \( b = -13 \), and \( c = -10 \) into the formula: \[x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4 \cdot 3 \cdot (-10)}}{2 \cdot 3}\]Calculate the discriminant: \( (-13)^2 - 4 \cdot 3 \cdot (-10) = 169 + 120 = 289 \). Solve: \[x = \frac{13 \pm \sqrt{289}}{6} = \frac{13 \pm 17}{6}\]This gives the solutions \( x = 5 \) and \( x = -\frac{2}{3} \). Therefore, the x-intercepts are \( (5, 0) \) and \( \left(-\frac{2}{3}, 0\right) \).
06

Sketch the graph

Plot the vertex at \( \left( \frac{13}{6}, -\frac{149}{12} \right) \), the y-intercept at \( (0, -10) \), and the x-intercepts at \( (5, 0) \) and \( \left( -\frac{2}{3}, 0 \right) \). Draw a parabola that opens upward, passing through these points.

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Key Concepts

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

Parabola
Understanding a parabola is crucial when working with quadratic functions. A parabola is a U-shaped curve that can either open upwards or downwards, depending on the leading coefficient of the quadratic function. In the function \( f(x) = 3x^2 - 13x - 10 \), the coefficient \( a \) is positive, meaning the parabola opens upwards.
  • The vertex is the point where the parabola changes direction.
  • The vertex can be a peak (if it opens downward) or a bottom (if it opens upward).
  • This graph is symmetric around its vertex.
It's comforting to remember that no matter the complexity of the equation, the graph of a quadratic function will always be a parabola.
Quadratic Function
At the heart of our discussion is the quadratic function, typically expressed in standard form as \( ax^2 + bx + c \). Each term鈥� \( ax^2 \), \( bx \), and \( c \)鈥攑lays a distinct role.
  • \( a \) determines if the parabola opens upwards or downwards.
  • \( b \) and \( c \) determine the location of the vertex and intercepts.
The vertex formula, \( h = -\frac{b}{2a} \), helps us locate the vertex horizontally, while substituting \( h \) in \( f(x) \) to find \( k \) gives its vertical position. Effortlessly applying these steps, we can decode the function \( 3x^2 - 13x - 10 \), finding a vertex at \( \left( \frac{13}{6}, -\frac{149}{12} \right) \). This point is pivotal in shaping the graph.
x-intercepts
The x-intercepts of a quadratic function are points where the graph crosses the x-axis. At these points, the value of \( f(x) \) is zero, meaning they are solutions to \( 3x^2 - 13x - 10 = 0 \). We find them using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substituting \( a = 3 \), \( b = -13 \), and \( c = -10 \) into the formula gives us the x-intercepts \((5, 0)\) and \(\left(-\frac{2}{3}, 0\right)\).
  • These points are where the function equals zero.
  • They are crucial for graphing as they give you distinct and easily plotted points.
By knowing the x-intercepts, we can plot these essential crossing points on the x-axis, helping visualize the entire parabola.
y-intercepts
The y-intercept is where the graph intersects the y-axis. This happens when \( x = 0 \) in a quadratic function, making calculation straightforward. For \( f(x) = 3x^2 - 13x - 10 \), substituting \( x = 0 \) yields \( f(0) = -10 \), therefore the y-intercept is \((0, -10)\).
  • The y-intercept is particularly useful for sketching the graph, giving an initial point to start from.
  • It simplifies understanding where the graph aligns vertically.
Knowing the y-intercept helps establish a foundational point for your graph. This, along with the vertex and x-intercepts, provides a roadmap for sketching the entire parabola without complex calculations.

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Most popular questions from this chapter

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=-x^{2}+8 x-17 $$

Without solving, determine whether the solutions of each equation are real numbers or complex but not real numbers. See the Concept Check in this section. $$ (2 y-5)^{2}+7=3 $$

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=x^{2}+3 x-18 $$

Neglecting air resistance, the distance \(s(t)\) in feet traveled by a freely falling object is given by the function \(s(t)=16 t^{2},\) where \(t\) is time in seconds. Use this formula to solve Exercises 79 through \(82 .\) Round answers to two decimal places. The Burj Khalifa, the tallest building in the world, was completed in 2010 in Dubai. It is estimated to be 2717 feet tall. How long would it take an object to fall to the ground from the top of the building? (Source: Council on Tall Buildings and Urban Habitat)

Find the maximum or minimum value of each function. Approxi- mate to two decimal places. $$ f(x)=-1.9 x^{2}+5.6 x-2.7 $$
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