Problem 49 - Algebra 1

Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=4 x^{2}+8 x-3 $$

Step-by-Step Solution

Answer

The graph of the function \(y=4x^{2}+8x-3\) is a parabola opening upwards with the vertex at (-1,-7).

Step 1: Identify a, b and c

For the quadratic function \(y=ax^{2}+bx+c\), identify the values of a, b, and c. In the given equation \(y=4x^{2}+8x-3\), the values of a, b, and c are 4, 8, and -3 respectively.

Step 2: Calculate the vertex

Calculate the x-coordinate of the vertex using the formula \(-\frac{b}{2a}\). Substituting the values of b and a gives \(-\frac{8}{2*4}\) which simplifies to -1. Then substitute -1 into the original equation to find the y-coordinate of the vertex. The vertex is (-1,-7).

Step 3: Plot the vertex and sketch the graph

Plot the vertex (-1,-7) on a graph. As the coefficient of \(x^{2}\) is positive, the parabola opens upwards. Sketch the graph of the quadratic function keeping in mind the vertex and the shape of the parabola.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point on the graph where the curve changes direction. It is a very important feature of a parabola, especially in graphing.To find the vertex, we can use the formula for the x-coordinate: \[ x = -\frac{b}{2a}\]Here,

• *a* is the coefficient of the squared term,
• *b* is the coefficient of the linear term of a quadratic equation.

Once you calculate the x-coordinate using the formula, you plug this x-value back into the original quadratic equation to find the y-coordinate.This y-coordinate, together with the x-value, forms the vertex, expressed as \((x, y)\). In this exercise, the parabola's vertex is \((-1, -7)\). This vertex is a crucial part of sketching the graph, as it gives you a point where the curve starts opening. Since the term with *x²* is positive (4 in this case), the parabola opens upwards. Each parabola's vertex details the minimum or maximum point on the curve depending on its direction.

Quadratic Equation

Quadratic equations form the backbone of parabolas. Generally expressed as \[ y = ax^2 + bx + c \]where each letter represents:

• *a*, the coefficient of x², determines the width and direction of the parabola,
• *b*, the coefficient of x, impacts the position of the axis of symmetry,
• *c* is the constant term which affects the y-intercept.

The solutions of a quadratic equation give the points where the parabola intersects the x-axis, and can be found using methods like factoring, completing the square, or using the quadratic formula. Understanding each component helps in accurately drawing or analyzing the graph. These components guide the shape and positioning of the parabola on a graph.

Parabola Graph

A parabola graph is a visual representation of a quadratic equation. It's essential to understand that the parabola has a U-shape, generated by the graphing of quadratic functions. There are distinct features of a parabola which are useful in graphing:

• The **vertex**: The turning point of the graph.
• The **axis of symmetry**: A vertical line which the graph mirrors over, calculated using \[-\frac{b}{2a}\].
• Open direction: Determined by the sign of *a*. In our function, since the value of *a* (which is 4) is positive, the parabola opens upwards.

To plot a parabola, start by placing the vertex. For example, in this exercise, the vertex is at \((-1, -7)\). From there, use additional points around the vertex for precision, by choosing x-values and computing their y-values using the quadratic equation. Then, connect these points smoothly to form a U-shaped curve. This method ensures the accurate plotting of a quadratic equation on a Cartesian plane.